One for the tyre whisperers

Ok back story. One in particular of my renegade at5 has historically been a pig to balance, various people have tried. Anyway I found a place that actually got it right in about November time and all was good except that the only wheel in question was observed as having some noticeable run out on the balance machine and took a bit more weight than the others. So being well happy to not have steering vibration for once I put it on the back and ran it for a while. Cut to last weekend and while on a mission to Narnia aka finding the corners of my workshop floor I discovered that I have a 5th alloy wheel that had been buried and forgotten about. So I took it down to have it checked for run out and it's good and straight and round. Took it home and painted it to match and yesterday I had it built up. Here's the problem the wheel is straight but they tyre is not, it's better but not perfect. This was without it having been driven on assembled. Is it possible that the tyre could have aligned itself with the old wheel and that running it on the new one it will work it's way back to true or do I need to be thinking about another tyre which is obviously actually going to be a new pair for the front.