Since the GOP's attempt to repeal and replace Obamacare with their god awful bill failed, I think that the next Demcratic wave won't push for single payer but will instead try to transition the us healthcare system into a multi payer system like Germany or the Netherlands. Now if the GOP healthcare plan had actually passed, then single payer would almost certainly be on the Democratic Party platform going into the 2020's.
I have no doubt the Democrats would be stupid enough to do this. Pushing a multi-payer system is a horrible idea politically. You have to realize that in every country that has adopted single payer, the healthcare debate is over. Once you have single payer, that's it, everyone realizes it's good. You might have a few people on the libertarian right who want to turn back time but no one takes them seriously. Anything short of single payer just prolongs the debate. Just get it over with and have single payer. Plus multi-payer is way harder to sell because it's way more complicated. Single payer is very easy "You pay taxes and then you get stuff for free".
By the way, Henry Jackson supported single payer.