Peter,
Was his residence at the time of enlistment Texas? I'm not too familiar with there records, but whereever he resided may have a probate, chancery, or land record in which his death could be recorded. Even if he died intestate (without a will), often these deaths are recorded in the County (or State Archives--depending on the State). I'd start with the county that he was living in at the time of enlistment, then proceed from there.
Ann