The interaction is ambiguous, so the DM must make a choice
When you employ an effect like the 6th-level Sor/Wiz spell magic jar [necromancy] (Player's Handbook 250–1), and you successfully attack the soul of a nearby foe, you possess that foe's body. That means you "keep your Intelligence, Wisdom, Charisma, level, class, base attack bonus, base save bonuses, alignment, and mental abilities[, and the possessed] body retains [i.e. grants you] its Strength, Dexterity, Constitution, hit points, natural abilities, and automatic abilities."
As the question says, the magic jar spell doesn't mention anything about creature type and subtypes, but this DM recommends a house rule like If a creature's mental abilities move to a new body, its type and subtypes are as per its new body until its mental abilities leave the body. Afterward, everything's back to normal.
This is how I've experienced magic jar effects the few times they've been used in my 20 years of play. To this player and DM, this has always made sense: creature type and subtypes just don't seem to me like mental abilities. In fact, I always thought that many magic jar users would aim specifically for sentient undead and sentient constructs (despite the latter usually being impossible targets) because of the immunities and resistances granted by those creature types.1
To put this in the context of the rules, this answer traces in greater detail the term normal form (and its synonyms natural form and original form) over several texts, but relevant here is this definition of normal form: Your normal form is you absent all effects that change your form. In context, though, this definition usually applies to a creature's physical form then usually only to comprehensive and full-scale changes, like a human changing form into a horse.
But when you use a magic jar effect, you aren't changing your form. Instead, your immaterial brainy-souly bits typically trade places with your foe's immaterial brainy-souly bits. Hardware's the same; software's different. (Your own analogy may be more precise.) This leads me to believe that the unit that's [your brainy-souly bits] + [a possessed body] is your normal form until your brainy-souly bits occupy another newer form or the magic jar effect ends.2
With all that in mind (ahem), the 2nd-level Sor/Wiz spell alter self [trans] (PH 197) when cast by creature that has successfully possessed a foe through a magic jar spell can be adjudicated with relative ease: "You assume the form of a creature of the same type as your normal form (such as humanoid or magical beast)"—or whatever the the possessed body's creature type is. In the case of our human wizard who has possessed a sentient zombie hill giant, that means a creature with the type undead. This reading meets this reader's expectations: it's easy, playable, and—to this longtime player—reasonable.3
If type and subtypes do move with the mind
Things get tricky if the DM rules, for instance, that type and subtypes are mental abilities. I can't know exactly what sort of house rule a DM may institute to dispel this ambiguity, but I can imagine a DM making a house rule like A creature's mental abilities move to a new body, its type and subtypes are as per the body that created the effect that allows its mental abilities to move. While this may work in theory, in practice this gets strange. Sure, the human wizard can't benefit as much from a magic jar spell that's used to possess a stone golem (possible otherwise under laboratory conditions), but now the human wizard can also drown if she possesses a kraken (it lost the aquatic subtype) or experience the corporeal world normally if she possesses a shadow (it lost the incorporeal subtype).4 I suspect this house rule will cascade into a list of exceptions and more house rules, culminating with a warforged wizard PC possessing Orcus and the DM rending his garments.
Nonetheless, if this is the house rule, then a human wizard who possesses a sentient zombie hill giant can still use alter self only to assume, for example, the form of a phaerlock (Underdark 97) or another creature with the humanoid type. While that strikes me as odd, if that works for you, don't let me yuck your yum.
If normal form means the caster's unconscious body
If the DM rules that while a creature is possessing another using a magic jar effect that her normal form is her coup de grace-ready body over there, then things go from tricky to weird. Such a house rule might read If a creature's mental abilities move to a different body, its normal form is the body that created the effect that allows its mental abilities to move. Or something? I mean, I don't know because I've never considered using this kind of house rule. That's not to say it's bad; I've just never seen this as a possibility.
This house rule means if you magic jar into a sentient zombie hill giant and cast alter self then you can transform only your unconscious body into a phaerlock. Yay? Anyway, that sounds to this player like code for the DM ruling that folks who can use magic jar effects can't themselves change shape while using a magic jar effect. I suppose that's an okay house rule? I mean, certainly, if the DM's had problems with magic jar effects before, the DM should make whatever rule allows the campaign to continue, but I suspect that rule will impact NPCs way more than PCs. Anyway, this player wouldn't leave a campaign over that house rule, but I'd still want to know it before the campaign began.
1 Full Disclosure: Although I've read the Wild Cards series that includes the Jumpers, I've never run or played a campaign wherein mind-switching was a central feature; I look forward to answers or comments with more firsthand experience. In fact, having PCs who can switch bodies is something I actively discourage among players in my campaigns because it's often unbalanced.
2 The magic jar spell isn't a spell of the transmutation school so it shouldn't be making any wholesale changes, but spells of the necromancy school are an eclectic lot. Still, monsters that use magic jar effects tend to "vanish" or "merge" into their hosts (Monster Manual 118 and Monster Manual II 201, respectively) rather than changing the host, offering possible examples as to how the effect could work.
3 Technically, this house rule breaks down a little with the type animal in that "no creature with an Intelligence score of 3 or higher can be an animal" (MM 306); the DM could rule that using magic jar to possess an animal makes the possessed animal's type and subtypes temporarily magical beast (augmented animal), but the magical beast type means that the animal-with-a-human-soul gains darkvision and loses the ability to be targeted by spells that only affect animals, and both of those are weird. It's easiest just to treat magic jar effects as an exception to the rule that animals can't have higher than 2 Intelligence, like one does when one plunks a headband of intellect +2 on Fido. Y'know, as one must at least once per campaign just to see what happens.
4 For a relatively small fee, a creator can imbue a stone golem with rudimentary intelligence (Dragon #327 73). Remove such a golem's magic immunity by having a friend hit it with stone to flesh. Remove its immunity to necromancy effects by, in the same round, having another friend hit it with the spell greater humanoid essence (Races of Eberron 186). Then you can finally try your magic jar spell against the stone golem—in the same round, so probably once. Maybe two 6th-level spells and one 7th-level spell might be worth a chance at a couple of hours as a stonebot gladiator in the right campaign? I dunno.
Note: For comparison, the Expanded Psionics Handbook includes what can be read as the psionic version of the spell magic jar in its mind switch power. The power's description includes this line: "You gain the type of your assumed body." Arguably, as a subtype is normally just a "subdivision of creature type" (see PH 313), this can be read as including the assumed body's subtypes by default. Ask the DM.
Finally, when I use the term house rules I'm not saying that someone's doing something wrong or that somebody can't read, and I never use the term house rules pejoratively or to minimize. At its most basic, whenever the DM finds a hole in the rules and fills it so that the game can continue, the DM's made a house rule. Such is the use of the term here.