  In the context of the Einstein-scalar-Gauss-Bonnet theory, with a general
coupling function between the scalar field and the quadratic Gauss-Bonnet term,
we investigate the existence of regular black-hole solutions with scalar hair.
Based on a previous theoretical analysis, that studied the evasion of the old
and novel no-hair theorems, we consider a variety of forms for the coupling
function (exponential, even and odd polynomial, inverse polynomial, and
logarithmic) that, in conjunction with the profile of the scalar field, satisfy
a basic constraint. Our numerical analysis then always leads to families of
regular, asymptotically-flat black-hole solutions with non-trivial scalar hair.
The solution for the scalar field and the profile of the corresponding
energy-momentum tensor, depending on the value of the coupling constant, may
exhibit a non-monotonic behaviour, an unusual feature that highlights the
limitations of the existing no-hair theorems. We also determine and study in
detail the scalar charge, horizon area and entropy of our solutions.
