Thanks! Powell’s hybrid method can be implemented with different variations, but I found this description for the MINPACK version GNU Scientific Library — GSL 2.7 documentation
which provided the details I wanted. In a summary " The Hybrid algorithm retains the fast convergence of Newton’s method but will also reduce the residual when Newton’s method is unreliable." Thus, it should be useful for the case I mentioned, and would be also safe with non-log-concave likelihoods (which may have other problems with Gaussian approximation, but that’s another story).
Aki