Give up the small point on purpose. It makes you look reasonable when you hold firm on the big one.

Conceding something visible early changes how everything after it is heard. Once you have agreed with part of what the other person said, you are no longer someone who just disagrees, so the thing you hold on to reads as principle rather than stubbornness. Winning every point makes you look impossible, and impossible people get ignored rather than answered.

Before your next difficult conversation, decide in advance which part you genuinely do not mind losing. Give it away early and say so out loud. Then hold the one thing that matters without raising your voice, and notice how much less resistance it meets.