Full size? Depending on hand size and arm length, a full 34″ scale length can be a bit of a stretch for some. There are some shorter scale instruments about – Gibson and hence Epiphone usually tend to be 32″ scale, and some of the “student” model Fenders (e.g. Mustang) are 30″. Slightly “out there” suggestion, but if scale length is an issue, a 5-string might be worth a look (play everything one string further down but 5 frets higher up).
Also consider neck shape. As a general rule, a Fender P-bass (Precision Bass) tends to be a not very deep front to back but wide side to side (which can be handy if you tend towards the sausage-fingered), whereas Fender J-bass (Jazz Bass) tends to be comparatively deep front to back but noticeably narrower side to side (strings closer together, so encourages accurate fingering). It’s a personal preference, but I generally prefer a deeper narrower neck on a bass.
The right instrument is the one that the player feels most at home with, as it’s the instrument that’ll get picked up most frequently. So go into a store, try a bunch and see what you get on with.