There are ways to temporarily increase your mining strength, if you’re right on the edge of being able to dig through a certain type of wall. Enhancing your tool with the repair station, certain clothing, and I think certain food recipes as well.
But sometimes the answer is just to go somewhere else to find new ore for the next stage.
I built my first cart track today. It makes it much quicker to check on my automated mining spots, but I have plenty of resources at the moment. I just need to track down more of the bosses.
I thought it would be atmospheric to build a graveyard out of mossy ground. But that just made the associated enemies appear so I dug it up and used a more neutral type of ground instead.