You could use an assembler like IDBA-UD, Spades or Ray (and there are others as well of course) to do de novo assembly in order to "build new genomes" as you write. You may end up with both old and new bacterial contigs, host genome (yak) contigs and perhaps some viruses. Of course you could also try to map your unmapped reads to the host genome to remove those from the analysis, and then map the remaining reads directly to a viral reference.

After assembling your unmapped reads, you can BLAST them to reference genome. It increases mapping rate.