This sounds to me like artificially creating an election-wide multi-member district. I don't think it's standard to allow votes outside of a particular voting district to affect the results in that voting district.

I'm not claiming that making districts multi-member is a solution (as you mention, multi-member, winner-takes-all districts are also bad), just that it is a necessary condition to arrive at results that are closer to representing the choice of the voters.

Edit: apologies if this feels like moving the goal-post. I need to think a bit more about what could happen for example in single-member voting district with ranked-choice voting system - I think it would not lead to a 2-party system, so you're right here and changing the voting system may be sufficient.