89 Toyota Supra N/A won't crank

[pica]

I have been all over the net looking for help with this problem. I bought an 89 Toyota Supra off a guy last year that had 130k miles for $500 with minor front end damage (1 headlight won't go down due to the corner being pushed in). Had to do some minor maintenance, (water pump, soft plug, hose on back of motor, tranny filter) nothing too terrible. When I first got the car he said I might occassionally have problems starting it. He said he always hooked up jumper cables or a battery charger and as soon as he hooked up the alternate power source it would fire right up. At first it was only if it rained the night before or if it was real humid out you'd have to turn the key about 2-3 times and then it would crank and start right up. So I went months without any issues. The problem has recently been getting worse, sometimes I just can't get it to start, come back 20 min later it fires right up. I tried the jumper cables trick and that did work fine twice, now that doesn't even work. When I turn the key I hear the starter click, but that's it. Funny thing is if I run it for 10 min at least, then i can shut it off and start it back up the rest of the day unless I let it sit for a few hours and then I have to go through all of it again. Toyota doesn't know what it is, they have suggested replacing the starter. Other sites i've seen say it may be the security system, however when that is engaged the starter doesn't even click, it just does nothing. Some say it's the starter relay. Im curious if anyone else has had these problems or what they might suggest to fix it. Also I was going to pull the starter to have it tested, but I can't for the life of me figure out how to get it out. One bolt I took out from the bottom. But the other bolt looks like it's behind the engine with no viable way to reach it. Any help would be really appreciated on either of these matters. Thanks