1989 Toyota Corolla intermittent starting problems

Hi and thanks in advance for the advice!

My 1989 Toyota Corolla GT-S (manual transmission) has an intermittent problem with starting: when I turn the key in the ignition, there is no click, no laboring of starter, just a low sound like a fan coming on somewhere in the engine. The lights and radio work. If I push-start or roll-start the car, it starts, then for the rest of the day, it starts up fine, but if it sits overnight it will not start in the morning.

The starter was replaced 400 miles ago: the problem has been intermittent since then, but at first it only happened once a month- now it happens every time I park it over night.

Is it the starter, the batter, the alternator? Help!

Thanks!

-Laura