Sounds like you still have a partial Blockage.
A lot of times there is either a
Q-tip OR a wire from deodorant that hangs on the side of the toilet and they fall in. using a regular snake the head is too small and ends up passing it (path of least resistance) TRY using a Drop Head Auger (WC SNAKE)
The older toilets do require a lot of volume for a proper flush so make sure the tank bulb is not dropping prematurely
You can adjust the lift rod and guide rod for a longer flushing action. These things are a real pain sometimes just hang in there .