I do not know any special trick to get it to advance beyond the "TEST" step or mode. Like you said (and on my car) additional button pushes will advance the odo from TEST to the other test modes or test parameters. Not sure how many there are, but I would guess there are about 35 button pushes/steps before it will loop-back and start from the beginning. Maybe make your button pushes a little more firm or a little slower? As I recall, there is one step that hangs up a little on my car, maybe it is the same one you are describing.
Check for DTC's (codes) using a OBD scan tool and not using the odometer test mode. Car part stores will often scan your car for free.