I am not interested in read mapping *per se*.

I am interested in reporting **ALL** possible end-to-end alignments of a read against a reference that have <= k mismatches (e.g. k = 5). I do not care about the "best" alignment.

Standard DNA & RNA read aligners (bwa, bowtie, STAR, etc.) that try to find the "best" alignment for a read place limits on number of mapping locations for short seeds, or something similar.

Does any read aligner allow FULL sensitivity?

For example, I can use MUMmer3 to report **ALL** exact matches of a read in a genome (of a given substring length). I am looking for something similar - i.e. full sensitivity - but allowing a limited number of mismatches.

Anybody know of a program that does this? I have been looking but am not convinced of the full sensitivity of the aligners.
There are several "all" mappers, [segemehl][1] is one I see recommended frequently. Also, see [this][2] (long but even so incomplete) list, search for "A" under "alignment reported" for all-mappers. Finally, if you are wary of results, create a small test data set and benchmark the tools.

 [1]: http://www.bioinf.uni-leipzig.de/Software/segemehl/
 [2]: http://www.ebi.ac.uk/~nf/hts_mappers/