Validity of worrying

I wanted to ask. I try expeditions now and am afraid of my rivals while seeking the tomb, since I can only afford 1 action-1 progress. But should I worry about them even then?

I don’t think so. The only punishment for having them beat you is having to repeat the expedition. You won’t, for example, miss out on the Tomb of Seven forever just because Virginia beat you to it. I’ve been beaten several times, and it’s no big deal, as long as you can tolerate the grind for more Supply Crates.

If you want help increasing your Watchful or gaining Sudden Insights, I’d be happy to play games of chess with you. I’m currently using chess games to increase Watchful to 200, and I’m always happy to find a new partner. :)

If you bring extra supplies, you can also find ways to sabotage your rivals during the expedition, or confront your rivals if they reach the Tomb first. The wiki has details on what you need for each option.

If your watchful is low, bring extra supplies. That helps, you can take advantage of the options to hinder your rivals. (And if you don’t use the extra supplies, you keep them, the expedition doesn’t automatically use them all up.)

Truth is, sometimes an expedition is just jinxed. The rivals stride along regardless of whether you’ve been successful or not. And they only ever have 10 steps where you might have 30. Or more! And nary a sign in sight.
I remember doing the Tomb of Seven with a vast pile of supplies and just scraping home with two or three left. Even now, I always take an extra 10 to be able to confront whoever is poking their nose in.
So as Optimatum said, extra supplies is the way to go.

For the really long expeditions, confronting them is almost required. As others said, bring along extra supplies and you’ll be able to persuade them to go away and reset your rival’s progress.