starts after setting overnight but then doesn't start again after driving. died in reverse also

Always starts after sitting a long time. Last 2 weeks it won't start after I drive it and come back out to leave. Sometimes I have been in shopping an hour, other times, just a short stop. Sometimes a 30 min drive, other times close to home.
After it sits a long time again, it will finally start.
Last 4 times it died, it started with jumper cables hooked up to it, so I went ahead and put in a new battery. However, both the battery and altenator had tested ok! 2 times, after it idoled for just a bit, and then was put into reverse, it died and would not restart.
Since the new battery was installed... I can only assume it did not fix the problem because 1 of the 2 "reverse dies" just happened to me again!
Please help!
After it sits a long time again, it will finally start.
Last 4 times it died, it started with jumper cables hooked up to it, so I went ahead and put in a new battery. However, both the battery and altenator had tested ok! 2 times, after it idoled for just a bit, and then was put into reverse, it died and would not restart.
Since the new battery was installed... I can only assume it did not fix the problem because 1 of the 2 "reverse dies" just happened to me again!
Please help!
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