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.
^^^ Yes, you need to have any engine codes "scanned' through the OBD port - inst. cluster testing will only show codes for the cluster.
To enter the test mode for other functions than engine codes, you need to hold the reset from car off to started - running - and showing "test" before releasing it. Just pressing the button with the car on will NOT get it into the test mode.