Short answer: Texas law requires a 31-day grace period for every premium after the first. Death benefit coverage continues during the grace period unless the policyholder gives the insurer written notice of discontinuance.
Under the Texas Insurance Code, a group life policy must give the policyholder or premium payor a grace period of 31 days for any premium other than the first. During that time the death benefit stays in force unless the policyholder notifies the insurer in writing that it is discontinuing the policy, as the policy provides.
The policy may make the policyholder liable for a pro rata premium for the time the coverage was in force during the grace period. So a death during the grace period is covered, but the employer can owe the premium for those days even if it then lets the policy lapse.