And now to do some redistricting specifically for this...
Speaking of which, the more districts you have, the harder it is to gerrymander. Added bonuses are that lobbying is more expensive this way and it is cheaper for candidates to run.
The New Hampshire State House does this very well. I can't think of any downsides to this idea.
The number of seats in the US House you’d need to eliminate or severely limit the possibility of gerrymandering is -much- larger than any serious proposal to expand the House. Consider state legislature seats in swing states like PA or WI. They are more than 10x smaller than congressional seats, yet those bodies have been far more effectively gerrymandered than their congressional districts ever were.