What do you do when there's no solid spot left to bolt down a core drill?

I've been in that exact spot more times than I'd like to admit, and a shimmed base with zero rock is fine for finishing out a handful of holes. The washers are doing the same job a wedge or a slurry of fast-set patch would do, just faster. The risk isn't the shim itself, it's wheather the shim can walk or crush under load. Flat washers tapped in tight between two hard surfaces don't relly have anywhere to go, so if you can lean on the base and it doesn't budge, you're good. One thing I'd do is check them again after each hole, because the vibration can loosen a shim that felt dead solid when you first pounded it in. Keep a hammer within reach and re-tap before every start.

As for the strip between your existing holes, it's almost certainly sound. Core drilling doesn't micro-crack the surrounding concrete the way people assume. A diamond bit cuts, it doesn't bash, so the material between holes is typically as strong as it ever was. What actually compromises those narrow strips is hitting rebar or aggregate and levering the bit around, or if the original holes chipped out badly at the edges. Take a look at the hole edges. If they're clean and crisp, your strip is fine. If they're all spalled and ragged, be more careful.

The bigger point on the bit grabbing is that bit grab ussually comes from the drill wandering on startup or from letting the bit spin down while it's still in the cut, not from a slightly imperfect mount. Ease into each hole, keep steady even water flow, and back the bit out a couple times on deeper passes to clear the slurry. That matters more than whether your base is on virgin concrete.

If you ever run out of flat surface entirely, the classic move is anchoring a piece of angle iron or a 2x across the area with a few Tapcons and bolting the base to that. But for what you've got left to do, your washer trick will hold. Just keep an eye on it between holes.

One thing that helped me when I was coring through my garage slab a few years back was scribing the outline of the base plate onto the concrete with a pencil before tightening down, so between holes I could tell at a glance whether the whole assembly had crept even a fraction of an inch. Also, if your stack is more than one washer thick, consider swapping to a single flat piece of steel shim stock or even a scrap of hardie board cut to fit, because a pile of washers acts like a little pivot and can let the base tilt microscopically under torque even when it feels solid under hand pressure. Pedantic maybe, but it bought me peace of mind.

Similar thing happened to me coring a wall for a mini split line set years back, and what saved me was cutting a scrap of 3/4 ply with a hole just big enough for the bit, then anchoring that board down with a couple of wedge anchors in fresh spots. Base bolted to the board, board bridged over all my junky old holes, dead solid. Pull it off after and you're left with only the anchors to deal with. I still keep that same plywood jig hanging in the garage and grab it any time the mounting area looks questionable.

I'll add that a metal shim stack beats anything wood or plastic for this. I cored nine holes through a block wall for a generator install two summers ago and ended up stacking three washers under one corner of the base. Held dead solid the whole run, never re-tightened once. Steel on steel doesn't crush like a wood scrap will after a few holes of vibration. If you've got a bit of flat bar or an old floor plate lying around, that beats a pile of washers.

One thing to check is whether that wall is poured or block. If it's block, that strip between holes might be over a web or a void, and your base can feel tight on the face shell while sitting on nothing behind it. Tap the strip with a hammer and listen. Solid sounds dull and flat, hollow rings.

Update: the ply bridge idea worked way better than my washer shims. Cut a scrap with a hole for the bit, anchored it down on fresh spots, and the base is dead solid now. What I'm still stuck on is that narrow strip between my old holes. Any way to tell if it's actually cracked up in there before I trust it, or do I just eyeball it and go?