The original midimuso is broken by design; see discussion here:
TL;DR the LM317 isn’t guaranteed to work with 12 V as used in that circuit, and odds are it’s not even regulating; you’re just seeing the supply voltage minus a collector-emitter voltage drop across the internal darlington transistor.
You may get around it by trying different LM317s, but the only robust approach is to increase the supply voltage (which means changing the supply if the one you have is too good).
In a quick test, I needed 12.25 V in to maintain regulation at 10.60 V out:
but the exact drop depends on the current; the datasheet suggests a 3 V difference for the regulator to work properly across the full range.