<p>I recently used BLAT to generate a large number of alignments but the input files were on the chromosome level, not individual genes. I realize BLAT and filtering the subject by taxonomical ID is an option, but BLAT seems to be a stricter option. The alignments are 60bp with no gaps. What would be the best way to find the gene associated with each of these regions, if it exists?</p>

As @AndreiR as said...

Download a bed file of all the genes of interest (from the ucsc table browser you can download ready-made bed files)

then make a bed file from your alignments:

then install <a href='https://code.google.com/p/bedtools/'>bedtools</a> and:

    intersectBed -a alignments.bed -b refseq.bed -wo > intersections.txt

This should give you all the info you need.