Wow, where do you live that your lilac has finished blooming already? Mine is just starting to show some color in the buds.
Your "zombie" lilac may not be dead, but personally I wouldn't think it worth trying to save, if it hasn't even leafed out when its neighbors are finished blooming. If it somehow pulled through, it will be stunted for years. And I probably would not put another lilac back in the same spot. I don't know what happened to it, but if it got diseased, there could be disease pathogens still in the soil. If not diseased, then maybe there is some combination of circumstances there that is not conducive to the lilac, maybe a low spot that holds too much moisture around the roots, not enough sun, some localized soil condition, etc. Unless you can figure out what it was and remedy it, don't put another one there.
Welcome to the forum!