GGG Controller experienced a loss of separation between two successive departures when the second cut off the first; claiming the Tower's failure to transfer the aircraft in a timely manner precluded corrective action.

Narrative

Aircraft X was called rolling off of runway 04 at TYR. The aircraft departed several minutes later and continued runway heading (standard assignment) until it reached 5 miles out. At that time TYR Tower called and said Aircraft Y was rolling. I informed them that Aircraft X had yet to call me. Shortly after that; Aircraft X called. I radar identified the aircraft and told it to proceed on course; which ended up being south/southwest bound. At that time; Aircraft Y was airborne and approximately 2 miles off the departure end of runway 04. TYR Tower calls me and says 'Aircraft Y's looking for Aircraft X; MC.' They didn't get an acknowledgement; they just said all that over the shout line; then hung up. Aircraft Y calls me approximately 2.5 miles off the runway end; climbing out of 2;600 MSL (too late for me to prevent the separation loss and way past the point the aircraft should have been switched by Tower). I radar identified Aircraft Y; told him to maintain 3;000' and immediately called traffic. Aircraft X had begun his turnout on course; which was a right turnout to the south. I'm unaware of the exactly how close the aircraft came to each other due to the extremely high level of traffic and no one to assist me. Aircraft Y was quickly told to make a left turn on course; and was climbed out of 3;000 shortly after the aircraft were no longer a factor for each other.Recommendation: TYR Tower Controllers never switched the first departing aircraft to me and only did so after I informed them it hadn't called me yet. Then; when the aircraft turns on course; they launch another one and didn't initiate any sort of vector or altitude restriction. They instead held onto the subsequent departing aircraft far too long for me to initiate any sort of control instruction to prevent the possible loss of separation. I've filed a safety report in the past complaining of poor communication from TYR Tower. They don't switch aircraft until it's too late to do anything to prevent dangerous situations (far past the approximately 1/2 mile off runway end). Whether it's due to inattentiveness; or on purpose; it is dangerous and continuously leads to dangerous situations. They launched Aircraft Y when they knew they had just switched Aircraft X and should have known the aircraft would be turning southwest bound on course. There is an apparent (severe) lack of team effort and communication on their part.As I suggested in my previous safety report; the LOA between TYR Tower and GGG Approach Control needs to be re-written. It should require them to assign beacon codes to all departing aircraft; regardless of whether they're IFR or VFR. GGG Approach needs to be responsible for sequencing VFR aircraft as well into TYR. We currently don't and TYR routinely ruins IFR sequences by cutting aircraft off with their VFR inbound. Perhaps; in light of the topic of this report; TYR Tower should be required to call GGG Approach for a release on all IFR aircraft.