First, if it is feasible to reach a point where atmospheric pressure is at least increased as compared to surface pressure, then it will be at that depth where less structural stress will be placed on artificial environments that are built on Mars for habitation by humans
Well in that case, stop digging.
The atmospheric pressure on Mars is ~ 1% that of Earth.
Wouldn't matter how far you dug down (of followed a cavern), the pressure wouldn't increase sufficiently to make up for the effort of going down.
The pressure differential isn't that big a problem and we don't have to pressurize the inside to the level of Earth at sea level, as we (and most living things) do fine at lower pressures, particularly if we simply increase the partial pressure of O2 a bit.