Toady has actually (and he rarely does this) deliberately prevented .01 worlds from being loaded into .02 (or was it .02 in .03..? check the devlogs... either way, it affects this situation).
You need to do a new worldgen. There may be a possibility to regenerate exactly the same world, by finding the randomness seeds and other settings used in your .01 one and explicitly using them in your .04 (or later?) one, but it would be starting from scratch anyway. If there's not been too much messing with the worldgen mechanics, though, you should at least get the same geography/geology and thus be able to redo your embark. But it would be from scratch. And it might be anything from slightly different to completely so, depending on changes of how the data is used. Or changes to the reject/keep thresholds.
In short, might be easiest just to go for a new world, new fort. Grieve over the civilisation you've now left in stasis (or come back to it and experiment when you want a largely self-sufficient fort to mess about with, say, mechanical logic constructions... without having to strike the earth and do all the other fiddly things all over again). I very rarely migrate old saves into new versions, anyway, so as to get the full benefit of various bugfixes that might involve the raws or incorrect tile flagging.