Troubleshooting assistance - No response when turning key

I'm a complete noob when it comes to vehicles. I went out to the garage last night to start my 1971 Duster 340. It started up fine, but stalled when I went to back out of the garage.

When I tried to restart, I got nothing. No clicks or sign that that anything was happening. I did have dashlights and radio, and my headlights worked, but nothing would even start the engine turning over. (I had the starter replaced a couple of months ago...and I don't think it would be the starter because I would usually hear a clicking sound).

Two minutes later after opening the hood and staring aimlessly, it started fine. I've done a couple of random starts and stops, and backed it out for a bit with no problem.

I'm afraid it will happen again when I go out, and I'm not sure what to look for to prevent or fix this. Thoughts?