Helicopter pilot reported not requesting a Special VFR clearance upon initial call to ATC to enter Class C airspace after the weather had gone below VFR conditions.

Narrative

Due to another helicopter inbound to the same HAA (Helicopter Air Ambulance) helipad I was occupying (ZZZ1); I had to hurry and leave medical crew to clear the pad and reposition to ZZZ. Weather was marginal as reported by ZZZ2 tower but above 1000/3. Enroute; the weather seemed lower towards ZZZ; I obtained ATIS V; reporting 6 mile vis and 900 BKN. I contacted ZZZ tower with request for landing at FBO and the ATIS id and I received the squawk code but I forgot to also request a SVFR clearance since it was controlled airspace with weather now below VMC at ZZZ. After my read-back and ZZZ verifying my position; to the best of my recollection; Tower stated something like 'cleared into the charlie with special' and I replied 'roger; inbound to [FBO] with victor'.In hindsight; my mistake was that I forgot to ask for the SVFR clearance on initial call-up when I called to establish communications with the controlling agency; ZZZ Tower. Due to the haste of clearing the HAA helipad; deteriorating weather conditions (which were getting close to company weather minimums); flying at a lower altitude than normal (between 400 - 500 feet) and scanning outside more intently to avoid a potential collision with antennas because of the lower altitude; ongoing radio traffic chatter; I believe I was getting task-saturated and time-compressed. Also; I haven't had to use SVFR clearance in over at least 10 years so it didn't come to mind.In hindsight; had I 'slowed down' and 'pre-planned' in my mind for the possibility of lower VFR weather conditions or even weather below VFR minimums as opposed to MVFR conditions when I lifted; that might have allowed me to recall the requirement for the SVFR clearance if weather was less than 1000/3 and request it accordingly upon my initial call to ZZZ tower.