Auto revers tape players usually detect the speed of rotation of the supply spool to detect when to reverse. It's possible that the tape adapter you have either does not turn the supply spool fast enough or with enough torque. It's also possible that your tape deck is slightly faulty, maybe a bit of fluff in the rotation sensor (usually a slotted wheel with LED and sensor on the other side).
Try reversing the tape adapter and using 'Side B' rather than Side A and see if it makes any difference.
Try reversing the tape adapter and using 'Side B' rather than Side A and see if it makes any difference.