" Although I already had my FAA paperwork, I would have needed a checkflight to take an a/c solo and with only one day free to fly, I was happy to have the Instructor along."
You cannot take the aeroplane solo without having first flown a US BFR. Having the paperwork is only the first step and a local checkout does not count. This is something that a few people fall foul of. Often the FBO does not know the full ins and outs of having a person turn up with an FAA piggy-back licence and they don't check. I was at a PFA talk night once when a guy was proudly showing slides of how he had flown all around the Caribbean in a rented aeroplane. Afterwards I asked if he had a BFR and he said no, he didn't need one and got quite shirty with me when I told him afterwards his whole flight had been illegal. What he does like that is, of course, up to him but he was not insured and neither were his passengers so if he had hit the veg he could have been sued senseless/houseless/everythingless. I spoke with a guy at the FAA fairly recently and he said they were aware of the fact a lot of FBOs don't know the rules on this but, as a previous poster said, you get your Temporary Airman's Certificate (incidentally the FAA is not listed in the phone books under "F" but under Department of Transport) you then arrange your Biennial Flight review that should be an hour on the gruond and an hour in the air which may or may not include your local checkout flight to satisfy the FBOs currency and insurance regs.
THEN you can go solo and not before.
Hope that helps,