‘Supersymmetrizing’ an arbitrary quantum-mechanical potential

To my understanding, it is not possible to $``\text{supersymmetrize}"$ an arbitrary quantum-mechanical system unless one knows how to represent the corresponding Hamiltonian in the form $$ H = A^\dagger A \quad, $$ which can be as difficult as solving the Schrödinger equation. (I guess, in QM we can generate SUSY potentials by playing with arbitrary superpotentials $W(x)$.)

So, I realize that my question is not very rigorous. Anyways,

Is there ANY way to construct a supersymmetric Hamiltonian $H_S$ from a given one, $H$? (without the necessity to solve Schrödinger equation for $H$ or doing some equally hard things)

What I mean here by $``\text{construct}"$ is not necessarily what people typically mean by this word in SUSY QM (see e.g. here). The only thing I want is SOME SUSY potential whose construction would somehow involve using $H$.

P.S.

Note that we know how to $``\text{supersymmetrize}"$ nearly any field theory. This suggests that a similar thing should be possible in QM.