Ain't nobody got time fo dat!You need to pull the cement plug.
1. Dig post hole.
2. Put post in.
3. Use brace to insure it's vertical.
4. Pour in quickcrete and let it set over night.
You could reuse the existing plug but it'll never been secure enough.
But really, the whole fence is very old and I'm fine with a part being a bit structurally insecure because by the time it fails the whole thing will need to go. I'd just be doing it now to do it again in 5-10 years. It seems like there should be a middle ground between full replacement and nothing.