The Avery transfers aren't too bad from my experience; I used them when I was just making a few shirts for myself only a long while back. Certainly didn't feel especially thick, though the hand was a bit higher than you'll get with Transjet II. How much pressure did you use when applying them? Did you press down heavily with both hands, or just use it like a normal ironing job with one hand?
As far as re-pressing the shirts goes, I don't think it will probably help much. However, it also shouldn't hurt anything, so you can try it on a few (and yes, make sure you don't forget the teflon sheet) and see if it helps anything. With other types of transfers (e.g. Plastisol transfers) it can actually hurt to re-press them, but regular heat transfers should be fine to try it anyway. The instructons that came with the Avery paper might recommend a temperature already - if not, maybe 350-375 degrees or so for 10+ seconds (hard to say for sure since I've not used the Avery paper on a press & it is a re-press).
To sum up - go ahead and give it a try, but it might not help that much.