Hardware multipliers often use a sort of base-4-ish lookup table trick as well, using the Booth-Wallace algorithm. Booth's idea is to rewrite one of the inputs in base (usually) "4", except that the digits go from -2 to +2 instead of 0 to 3. (That's five possible digits! This helps the rewriting stage not have to propagate carries. Carry propagation is very expensive.) You can use Booth in a base higher than 4, especially if you know one of the multiplicands before the other, but you run into tradeoffs pretty quickly.

Then for each digit, you select between the other input multiplied by 0 (all zeros), +1 (identity), +2 (shift left by one bit), or -1 or -2 (flip all the bits of +1 or +2, plus a correction). Since a number has about half as many digits in base 4 as in base 2, you have about half as many digits to sum as if you'd done this in base 2.

Then you sum up all those results, but since carry propagation is expensive, you mostly use "compressors", e.g. you sum up three intermediates at a time, but you do it bit-by-bit, where three 1-bit numbers add up to a 2-bit number (from 0 to 3). This is called a Wallace Tree. The point is that you are generating carries, but you aren't propagating them, just adding them back into the set of things to be summed.

At the end of the tree step, you have just two numbers left, and you add them conventionally. That's the only step that needs full carry propagation.

If you are implementing a multiply-add, or multiplying several numbers and adding up all the results or similar, then you usually only need one full carry propagation stage.

The overall circuit has quadratic area but only a logarithmic depth in gates. IIRC whether to do Booth or not is a tradeoff: at least in some circumstances the rewrite steps make it slower but smaller. Hardware tool vendors have done a lot of work to tune these circuits very tightly, using e.g. specialized gates like AOI, heuristics for how to set up the tree, etc.