Q&A

I'm not sure what the humming sounds like, but listening closely could it be the starter motor? If this is the case, it means your trouble is in the drive connection between the starter and the ring gear on the flywheel. Normally this means the clutching mechanism is broken, worn, or gummed up (esp. aftermarket starters). Sometimes it wont get the starter's teeth to engage in your flywheel before the starter is turning too fast to let it. (safety to avoid tooth damage by grinding/jarring) FIXES: If that clutching mechanism is wearing, it will be an entirely intermittent problem. Replace the starter motor again. If its gumming up then it will usually react to heating. Working better in some temperatures than others. If sludge of some kind then it'll work in the heat of the afternoon, and worsen in the cold early mornings. You can remove the starter and clean it up as best you can focusing of removing all lubrication from that clutch, then use a proper lube for steel. Reinstall. If its worn metals and sticking it may actually work better in the cold, and really crummy in the heat. You may clean and lube but I'd recommend replacing. This is sometimes the case with aftermarket starters, especially 7 years old.

I had the exact same problem in my '96. It is because the contact plates in the starter are pitted and doesn't make a connection depending on what position it is resting in.
If you take out the starter you can replace just the contact plates. Toyota has a kit for this but so do many auto places (its 15 or 20 bucks). The plates are the flat looking copper parts mounted inside the starter. They are really easy to take out and put back in. If you go to Toyota, you need to take the plates with you because they have to match left or right handed plates and there is no way to tell without taking them out. I got a kit at Oreilly Auto parts that just came with the parts for both directions. It was really easy (aside from getting the starter out was a royal pain).