It's a little goofy, but the concept is solid...
This idea is based on the concept of line of effect for areas of effect:
A spell's effect expands in straight lines from the point of origin. If no unblocked straight line extends from the point of origin to a location within the area of effect, that location isn't included in the spell's area. To block one of these imaginary lines, an obstruction must provide total cover.
When the character is wearing the magically-reduced cone as a hat (it doesn't need to be lead) and they're not in an anti-magic field, the magic keeps it hat-sized.
When they step into the anti-magic field, the cone hat is within the area of effect of the anti-magic field, so the magic is suppressed and the cone returns to full size.
The cone now serves to block line of effect between the anti-magic field source and the cone-bearer, putting them in total cover. As indicated above, total cover blocks the effect. The cone-bearer would still be able to use magic, but he'd have to deal with the fact that everybody has total cover from him, too.
...but there's a catch.
Where do you get the hat? Enlarge/Reduce only lasts for a minute, requires concentration, and only changes the dimensions by 50%. It's not going to make a big enough change to the hat for this idea to work.
You're stepping into house rules territory for the hat itself, but a generic solid obstruction blocks fine.