The English have good records. We could perhaps find the decendents of relatives who stayed put and then find their "hey you guys seem to have more DNA in common than you ought to" counterparts of native american heritage.

The way it works for molecular phylogeny is that you try to find things that are conserved. E.g. if you find a small village in Europe where people haven't moved around much and you find a rare mutation that is also present in one other part of the US, then you might be able to put some numbers on the likelihood that this mutation/gene came from a the original place. Find a second gene, find some artefacts from the right place/time and you have an emerging picture.