St. Moritz

Let me guess. You are picking up the Glacier Express in Brig and taking it to St Moritz. Why else would you go so far out of your way to get to Innsbruck? I would just go to booking.com and look for something in your budget close to St Moritz train station. After all, you just need a bed for the night, and Swiss hotels have a very good standard, even the budget ones.

BTW, what is your route to Innsbruck? By rail to Buchs SG, then Railjet train to Innsbruck, or the "shortcut" with the train to Scuol-Tarasp, then bus to Landeck-Zams, then train to Innsbruck?

Or another shortcut: Train to Zernez, bus to Malles, then train to Bolzano and on to Innsbruck...

Lots of interesting possibilities there.

But anyway, you can also look at Pontresina or Samedan for Lodging. Does not have to be st. Moritz if just for one night...

I haven't spent the night in any of the towns along the Bernina Express route, but many sources say St. Moritz is not a particularly attractive place to stay unless you are skiing. Pontresina is often suggested as a nicer and less expensive option.