Get all local maxima of a DFT
That sounds a lot like you shouldn't be using the DFT, but rather a parametric spectrum estimator.

For example, there's spectrum estimators that actually give you a function to evaluate at real-valued points rather than a vector of a sampled spectrum (your sentence "DFT, unquantized" doesn't make all that sense to me – the DFT is discrete, inherently, so what you describe as search algorithm only works if you do a massive DFT and then start somewhere in the and look for peaks from these points on).

My go-to example here is either


ESPRIT, which, given knowledge about the number of significant signals will just give you a vector of real-valued frequencies of peaks or
ROOT-MUSIC (dunno if that's the official term, it's what we used, based on [1]) which uses the pseudospectrum function generated by the MUSIC algorithm and finds the signals by looking for roots of a specifically crafted polynomial [1].


[1]: Barabell, Arthur J.: Improving the resolution performance of eigenstructure-based direction-finding algorithms. In: Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP ’83., Band 8, April 1983, available online.