Is there a way to have a reach longer than 5 feet when not attacking?

Question

One reason you might want to increase your reach is when helping somebody escape from a net. The Net's description states:

A creature can use its action to make a DC 10 Strength check, freeing itself or another creature within its reach on a success.

Every way I have found of increasing your reach only increases it for attacks that you make: The Reach weapon property, the bugbear's long-limbed, the Battle Master Fighter's Reaching Attack Maneuver, and the Four Elements Monk's Fangs of the Fire Snake feature.
Even the section on "Melee Attacks" states:

Most creatures have a 5-foot reach and can thus attack targets within 5 feet of them when making a melee attack. Certain creatures (typically those larger than Medium) have melee attacks with a greater reach than 5 feet, as noted in their descriptions.

Which, to me, seems like it only applies to attacks and not reach in general.
Is there a way to increase your reach when you aren't attacking, such as when you are helping a creature escape from a net?
Both Hunting Traps and Quicksand also include the phrase "a creature within your reach", so anything that helps with nets would help with those as well.

Answer 1

Per the PHB, p. 195, as quoted by you:

Most creatures have a 5-foot reach and can thus attack targets within 5 feet of them [...].

RAI, PCs have a 5-foot reach around them

Since the PHB explains the core rules for players to follow in game, it is safe to assume that "most creatures" includes Playing Characters, as creatures of Medium or Small size; realistically speaking, a regular-sized humanoid can somehow interact with things within 5 feet from them without moving, especially with a weapon or a tool.

RAW, there is no specific mentioning of a character's reach

That is the only paragraph where reach by itself is explained in the core rules and it's paired with the concept of attacking. It is never included in a creature's characteristics, like Size or Speed, neither in the DMG or in the MM. Indeed, reach is regularly mentioned in the description of attacks, e.g., a Wolf's Bite:

Melee Weapon Attack: +4 to hit, reach 5 ft., one target

There's a correlated concept mentioned in the Space section of PHB (chapter 9, p. 191, emphasis mine), about the "cells" occupied by a creature:

A creature's space is the area in feet that it effectively controls in combat, not an expression of its physical dimensions. A typical Medium creature isn't 5 feet wide, for example, but it does control a space that wide.

Nothing else is said about what is around a creature's occupied space and their influence on said surroundings. This means that there are no detailed rules about creatures' reaches for interactions that are not attacks and the paragraph you quoted is the only generic guideline.

Since no specific ruling is given, the DM decides

The DM has to decide what can and cannot be done. When not attacking, does a Huge creature have, say, a 10-foot reach, thanks to their size, or just the "regular" 5-foot reach? Maybe that's the case for a Giant, thanks to their long arms. What about a Gargantuan one, does it have an even bigger reach? What about the Kraken, whose Tentacle attack has a 30-foot reach? Does that mean it can interact (without attacking) with things within 30 feet of it, thanks to its tentacles? Can a Dragon use their long tail to somehow interact with something out of the reach of their Claws or Bite?

If the DM rules that large creatures have a bigger reach thanks to their size, you could take advantage of this by Polymorphing, Shapechanging, etc., into a Huge creature, for example.

Regarding items, weapons with Reach, such as the Pike, add 5 feet to your regular reach when attacking with them (PHB p. 147, emphasis mine):

Reach. This weapon adds 5 feet to your reach when you attack with it.

However, you might be allowed to creatively use them for something other than attacking, in a plausible way. Referring to the examples you gave in your question body, helping someone getting out of a Net might require you to directly touch it, or somehow cut it from afar with your Halberd; you could help someone getting out of quicksand by reaching out to them with your Lance, increasing your reach beyond your arms.

Similarly, Bugbears, as a playable race in Volo's Guide, have a racial trait that grants a longer reach when attacking. Your DM might allow this to be extended to maneuvers other than attacking, if you play as a bugbear or are transformed into one.

Long Limbed. When you make a melee attack on your turn, your reach for it is 5 feet greater than normal.

Unfortunately, there is no spell that directly increases a creature's reach, in the PHB. As previously suggested, spells which increase size might indirectly increase reach as well, if the DM rules so.

Answer 2

Pet?

Alternatively, having a "pet" can effectively extend your reach, even if it isn't really your reach. This could be a ranger's companion animal, a trained pet, a familiar or even an intelligent mount who can follow instructions (like that from find steed).

Improvised Action with Reach Weapon

The Definition for Reach weapon property reads:

Reach. This weapon adds 5 feet to your reach when you attack with it, as well as when determining your reach for opportunity attacks with it.

Meaning the reach only applies while attacking or attempting opportunity attacks. However the rules also has rules to be purposefully open ended. An example is improvised action sidebar:

Your character can do things not covered by the actions in this section, such as breaking down doors, intimidating enemies, sensing weaknesses in magical defenses, or calling for a parley with a foe. The only limits to the actions you can attempt are your imagination and your character’s ability scores. See the descriptions of the ability scores in the Using Ability Scores section for inspiration as you improvise.

When you describe an action not detailed elsewhere in the rules, the DM tells you whether that action is possible and what kind of roll you need to make, if any, to determine success or failure.

"I want to use my reach weapon to snag the net and drag it closer to me" is a perfect example of an improvised action request.

The DM would determine the how of the thing.

• They might decide that, since grappling is an action, getting hold of the net with the weapon is a full action (maybe Dex check with disadvantage with DC of 14 or 16 [what I'd do]). And that you'd have to wait to attempt to pull on your next turn to attempt to pull with the weapon without just tearing a small hole in the net (strength check DC 12 or 14).
• Or they could say it is one action, one check.
• Or they could say "After considering it, you don't think you could catch the net with your great axe because it won't fit through the holes, and trying would likely hurt the person inside."
• Or they could say, "no."

Answer 3

Reach isn't something which can be applied. The description simply states that if you cannot reach the creature caught in the net, you cannot help it get out of the net. This is a situation where you step away from numbers and math and simply say to the DM "I try to free Jumbo from the net.".

I don't have a RAW answer to give to the current question. I will add that if I were the DM, I would allow any changes to melee attack reach based on your body to also apply for these situations. Eg. bugbears' Long Limbed¹ trait would apply, but not holding a glaive. However, in my eyes, this is all about using your hands to manipulate an object so it's more of a narrative question to the DM as opposed to a rules question.

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¹ Volo's Guide to Monsters, Monstrous PCs