I know this would only happen in an alternate universe, but they should be able to come out and say "ya know, we got it wrong" on certain issues, such as immigration. They will face a bit of immediate shame from pundits, but gain in the long term by removing that point of contention from the conservatives. Thus opening up a share of their voters.

To be fair to Republicans, they could say "ya know, we do believe human civilization has caused climate change and there is a government role to address it. We just disagree on the terms and mechanism for how that should work"