Basic survival advice is to stay put and wait for rescue unless you're in imminent danger. So maybe if your watch is your only lifeline stop running around the mountains once it dies.

Or use something intended for the purpose like a Garmin inReach. The satellite SOS features on phones and watches are nice to haves that could absolutely save your life but they're no replacement for being prepared. Really no device is a replacement for being properly prepared in the wilderness, even with your location you still need a way for rescue teams to spot you, for example.