Just thinking about the mechanics a bit, an idea of mechanical advantage comes to my mind. When the tape is about half used, the operating diameter of the "supply spool" is considerably less than it was when the supply reel was full. Since the tape transport speed should ideally be controlled at a constant rate by the capstan, that supply reel should need to turn faster in order to keep the tape moving at that constant speed. But what if for some reason there were extra friction in that supply reel's system.
Granted, I am not familiar with your tape deck; but I know that many - and perhaps most - decks have some kind of brake mechanism on the supply reel. On the Sony 377, for instance, the brake is "modulated" by pressure on a lever sticking up in the tape path just left of the erase head. As tape moves in play or record, this system achieves what I'll call a "dynamic equilibrium" in backtension on the tape by easing the brake just enough to maintain only a slight drag on the tape supply reel. If this brake mechanism gets out of adjustment, extra drag might occur as the amount of tape on the supply reel becomes less than half of the reel. I don't know what kind of "clutch" is present on the take-up reel of your deck. On older nonprofessional machines, this clutch might be a felt affair mounted between the take-up reel's platform and a spinning plate below. That system allows slippage so that the speed of the take-up reel can vary from its faster speed at the beginning to its slower speed near the end of the tape. Now if that kind of clutch is present, and if it has slightly too much drag, it might affect tape speed, particularly if there is only one motor for the whole system. If there is a capstan motor serving only for turning the capstan at a very precise speed, then I doubt that the take-up system is the culprit.
Just one more consideration, especially if there is a separate capstan motor: Question: Since the tape plays slower when it is more than half-way through, have you tried immediately mounting a different tape at its beginning and checking tape speed? I ask this question to rule out the possibility of a capstan motor becoming hot and tightening up enough in its bearing to slow the capstan speed slightly. Heat in that motor is not my first thought on your problem; but since I have worked on a few tape decks in my years, that is one problem which comes to my mind.
I hope the ideas from some of us on this list will help you fix your tape deck, and I wish you the best in your recording work. It is a lot of fun!