Instead of randomly digging hit the ground with the stag breaker and if you get the message "too hard" theres a good chance silver is under you. The "too hard" message can also come up if near obsidian or if you are over buried treasure of some type also found in mountains. Careful though as using stagbreaker attracts nearby enemies.

You can also get iron in the swamp this way and I often do it to get a stonecutter early on.