Melt and pour soap has solvents in it to allow it to melt easily. Regular cold process soap does not have these solvents, so it doesn't behave the same as M&P.
If you heat regular soap, it will soften to the point when you can stir it, but it stays fairly thick and gloppy. Most of us call this rebatching, not melting the soap. I'd agree that this is a melting process, but you'll confuse a lot of people if you use the word "melt" rather than "rebatch" when talking about regular soap, rather than M&P soap.
I've never met a rebatched soap that was better looking than the original, but YMMV. If you want to rebatch, do it soon after it's made. I would not cure the soap and then rebatch -- it can be done but it's much harder to get it all fluid. And you'll have to wait for another whole cure time to get the soap firmed up so it's nice to use again.