My first step would be what EXP100 suggests - do my own digging through the airline's route network and see if a creative routing is available. If it's an airline-caused delay (not weather), I'd also look at other airlines that I might be able to get the original airline to reticket me on.

Even if it's WX, I'd still look at other airlines and see if there's a reasonably-priced last-minute option. You can always get the original ticket credit refunded if a flight is canceled.

Fortunately, in the cases where there literally hasn't been another flight for a whole day or longer, I've been within 12 driving hours of my destination and no fast direct train option. So my call was easy at that point: get a rental car and go.

For an intercontinental itin where driving wasn't an option, I'd focus on getting my rebooking in hand as quickly as possible, even if it's 2-3 days in the future, and getting out of the airport. I would not try to hang around in hopes of standby unless I was top tier elite and had reason to believe I'd make the flight. Spending two unplanned days in, say, Paris isn't a terrible thing. Spending two extra days at CDG waiting on low-percentage standby opportunities sounds horrible.