Car randomly not starting, no cranking sound, had to jump start on 1995 Toyota Corolla

My car is about 260,000 miles. It never failed to start until six months ago, when the car randomly would not start with no cranking sound. Each time the jump start worked right away. I have brand new battery. In fact i thought the battery was the problem, so I got new battery three times the last six months. The not-starting problem happens once every few weeks, usually right after I had driven on freeway for about 20-40 minutes. I brought the car to the shop, the mechanics said the alternator is working. The starter might be on its way out. If it is the problem of the starter, why it started right away when jumped? Could it be the alternator not always charging battery when driving? I have both original starter and alternator. Thank you for any possible solutions.

Asked by for the 1995 Toyota Corolla
When it fails to start , what symptom does it give you? A simple 'click' , cranking but will not 'catch'(start), or no response at all from turning the ignition Answer EACH of those questions and help narrow down the cause(With headlights on and key in crank position , the lights don't change at all)? Answer EACH of those questions and help narrow down the cause.
When the problem happened, It had no cranking sound. The dashboard lid up, but no response from turning the ignition. This only happened five times for the last five months. I didn't try with the headlights on when it was not starting. If it happened again, I will make sure to test.
1 more answer , 2 more comments
do you have any eng codes?
My engine light has been on since 2008. Because it is a 95 Toyota, I am not able to get the code reading.
For code retrieval on that year , there is a diagnostic connector box under the hood (about an inch by an inch&a half) possibly by one of the strut towers. Open the lid and there should be a diagram of the terminals. With key off , put a jumper wire(Metal paper clip) into the ones labeled TE and TE1 , turn key to 'on' and count the flashes-ex.--code 34 would be 3 blinks followed by a short pause then 4 blinks , then a longer pause before the next code , if more than one.
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