Is this a non-model species for which there isn’t any known TCR loci information? If so presumably your worry is that those other species’ loci may be different enough such that you’ll miss some of genes?
As far as I can tell there’s two ways to solve this problem. One, the long but thorough way: sequence your organism’s genome/TCR loci and look for V/J genes. Two, do a 5’RACE experiment and just sequence whatever’s upstream of your constant region, finding V/J genes that way.
The second approach obviously only works if you know at least the constant region sequence(s), or near enough, and won’t give you much in the way of information about the locus/pseudogenes etc, but if all you care about is the expressed repertoire diversity then that hardly matters.
A nice tool called IMPre came out a little while back which sounds like it’s made to exactly that, although it hasn’t been used ‘in the wild’ on a wholly novel species yet to my knowledge. There’s also a pre-print out at the moment doing the same thing in bank voles), which seems to have used an unpublished tool called AmpliTCR for the equivalent step.