I start with digging out a hole to hide out initially underground. Put a floor/ceiling above level with the ground around the base area so that if any wandering hordes come by they walk right over. At the corners of the excavated area, I place my initial support pillars, and then determine where I want my 'dig' down point to be. I dig down a 1x2 hole and as I go down I make sure I'm crafting flagstone blocks and ladders, so that I can support the structure above with atleast one pillar to bedrock. The desert sand has enough SI load bearing to be able to handle a flagstone block stuck to it, and because it's a 1x2 hole, the flagstone blocks will be supported on three sides, so more than enough, even when digging down under your support pillar.
Alternatively dig a 1x1 hole straight down to bedrock, and then widen it to a 1x2 when you're at the bottom to start placing your support pillar and work your way up. Once you got a flagstone pillar to bedrock you can be sure that you have some SI load bearing at the top for the roof that is above at ground level.
When I built a hatch elevator, I added additional flagstone pillars to increase the overall support the base above had.
From there it's just a matter of knowing how far out from your central pillar your corner pillars are, so that if you hollow out the area down below, you can shore up the structure above at the X,Y coordinates down below to ensure you maintain SI all the way to bedrock. Never had a structure above collapse. With that said, I don't do much digging directly undernearth the base, and instead pick a direction to the nearest biome of interest, and start digging in that direction.
Unless you dig out a significantly large area, you shouldn't have too many problems with SI, since the blocks above your dug out area will get load bearing from the structurally sound blocks from the columns that you haven't dug out. The deeper you dig, the larger a cavern you can dig out, but regardless, it's always a good idea to put some support pillars periodically.
