IMHO, a lot of ideas could be more acceptable to the other side if “sold” in different packaging.
Single-payer healthcare: Saying “we need to do it out of compassion” is totally the wrong way, that doesn’t sell it. Rather, sell it to conservatives as a matter of national pride. This is one thing Trump could have pulled a “Only Nixon could go to China” on, but didn’t. Argue that “Americans deserve the best healthcare in the world, and what we have right now is a total ripoff.” Demand that the USA have as good, or better, a healthcare system as Canada, the UK, etc. Point out that other nations get better outcomes for less % of GDP spending on healthcare. It would still have been a tough sell, but someone like Trump just might have been able to pull it off - alongside of some vicious Trumpian attacks on the health insurance industry and then pointing out that single-payer is the same concept as health insurance itself, everyone pays into a pool and the pool then pays for everyone.
Abortion - rather than “a woman deserves the right to choose” - which totally backfires with conservatives - point out that if liberals are likelier to have abortions, they are aborting their future liberal voters and giving conservatives a future edge at the polls. (Whether this is actually true or not would need more study, but the concept is intuitive.) This could suddenly get a lot of support from conservatives for abortion that they wouldn’t have otherwise. (I don’t think this approach is ethical or good, but it might work - and this thread is only about what works.)
Free speech - rather than “racists, bigots and white supremacists deserve free speech” - which totally backfires with liberals - it should be pointed out that a conservative tyranny (like Trump is often accused of wanting or enacting) would crack down on liberal free speech (there is already an Atlantic article about this.) Furthermore, liberal ideas that were unpopular in their infancy, such as feminism or LGBT, could only have survived and thrived in a nation with a legal right to free speech.
Environmentalism - focus heavily on toxins. Leave the “we love nature and trees” out of it, that comes across as kumbaya-ish. Rather, focus all on the poisons being ingested. Conservatives don’t like the concept of mercury, lead, or arsenic in their food or water anymore than anyone else does. Hammering the toxic angle, and sounding the alarm on what toxins do when inhaled or ingested, could potentially sell environmentalism much better among conservatives than currently.