Seems some poeple are possibly missing the point here.
On completion of the JAA PPL, the licence takes a while to be issued so you have no licence to convert so if RuFer wants to start hours building straight away he can't do it on the basis of his validated as yet non-existant foreign licence. So the next best albeit not ideal thing is to continue to fly as an FAA student pilot as he will have a medical/student pilot ciertifcate.
However as stated previously any cross country flights he does will need to be authorised by an FAA instructor. Now unless things have changed recently, when training for a JAA PPL , solo flights still have to be authorised by an FAA instrctor so he/she would be the best person to continue to authorise further flights.
Now here's where to be careful. Watch for the guy who tells you during your JAA PPL training "We are training for a european licence, the FAA have no jurisdiction over what we do". This may include sending you on solos without the proper "authorised" sign-offs, sending you flying in aircraft that have exceeded their 100 hour inspection because you can use the 10 hour extension. Then I believe you will have found the real liar and he is out there.