Are you trying to align two mitochondrial genomes, next-gen DNAseq reads to a mt genome, or RNAseq to a mt genome? For DNAseq, BWA will be fine, for RNAseq, STAR or Subread should be fine. If you are building an index with just the mtDNA, you may have to adjust the index to account for the small genome. And if your reads are from whole cells / tissues, you will probably align NuMT reads to the mtDNA, this problem would probably be smaller if you align against the whole human genome and then extract reads mapping to the mtDNA.

edit: you may use GSNAP to specifically address mtDNA circularity:

Finally, you can provide information to gmap_build that certain
chromosomes are circular, with the -c or --circular flag.  The value
for these flags is a list of chromosomes, separated by commas.  The
gmap_build program will then allow GSNAP and GMAP to align reads
across the ends of the chromosome.  For example, the mitochondrial
genome in human beings is circular.

If you want to align reads to a genome, I suggest BBMap. If you want to align genomes to genomes, I suggest MUMmer - it's much better with very long sequences. Circularity is not really all that important; it just affects the tips.

CLC Genomics Workbench's read mapper works with circularized references and will map reads across origin without any issue.