TPA Controller reported missing a VFR traffic conflict due to workload resulted in a pilot reported NMAC.

Narrative

Aircraft X came over from F11 at 3000 ft.; requesting the RNAV Runway 28 at LAL via direct KONDE. I gave Aircraft X direct KONDE and had to descend Aircraft X to 2000 ft. to avoid VFR traffic opposite direction at 3000 ft. (not on my frequency). After that I cleared Aircraft X for the approach and issued traffic that was off his left that departed off a satellite airport (GIF). Then I called F11 to point out an aircraft who called for a pop up IFR off GIF and then issued the IFR to the other aircraft. Then I saw the traffic was passing behind Aircraft X at almost the same altitude and asked if they saw the traffic passing behind them. I thought it was the same traffic I had previously issued but then when I realized there was another aircraft out there; I apologized for not seeing the traffic sooner. The traffic that got close to Aircraft X was moving so quick I didn't not see the traffic as I was issuing the IFR to the other aircraft at the time. Aircraft X reported the traffic as a jet but never replied as to how close the traffic got. I later heard today that the pilot had called and is reporting it as a near midair.Recommendation: Possibly talk to local pilots that fly in/out this area. Develop alternate local recommended flight routes for VFRs to avoid traffic in and out of LAL. If we could make LAL a Class C airspace that would eliminate a lot of conflicts for arrivals at least on visual approaches and departures. A lot of the time you have to issue traffic alerts to these aircraft (climb/descend or turn).