93 Camry, clicks when turning key and then starts

I have a 93 Camry, 147,000 miles, V-6. The problem is when I turn the key to start, there's a click and then nothing. It resets the clock and the radio stations like the battery has been disconnected. I'll open the door and there's no ding indicating that the key's in the ignition. I pop the hood, shove the battery a little and I hear the ding and then it'll start. Sometimes all it takes is opening the door and waiting a second for the ding indicator and I'll try and it starts. Other times I just turn the key slowly and hold it like I'm starting it and after a few seconds it starts. This began happening last spring and I was told it was a bad battery and it was replaced along with a corroded connector. It's still happening. I can go for days and it's fine. And then it'll happen for 2 or 3 days in a row, dry or rainy weather. Is this an electrical problem or something with the ignition system? Any suggestions would be appreciated!