(edit was typing this out before you posted your rather AA's but there is one AA sugestion in there)
Hello and welcome
Ive noticed XR-E Q5 and R2 emitters are more of throwers due to the emitters viewing angle of 90 degree's and built in reflectors on the emitters so they put more light forward.
Where XM-L, XP-G emitters have a viewing angle of 125 degree's and put more light out the side then the Q5's and R2's, but because the XM-L, XP-G are usually driven harder then the XR-E Q5 and R2 emitters they put out good flood and throw. The T6 XM-L has to be one of the most popular most used emitter around today.
Just keep in very mind small XML lights like the Xeno E03, UF-2100 will get hot very quickly on lion rechargeable and chew through AA or 14500 batteries quickly, so they might not be able to be used for long times, due to heat and battery life.
Lights i have personal used in your price range that can be put in pockets.
Uniquefire UF-2100 T6 1-Mode Memory LED Flashlight (18650)
Maybe a bit big for the pocket, but a good light.
Trustfire C8 Stainless Steel Head XM-L T6 900-Lumen 5-Mode LED Flashlight (1*18650)
Ive got this light on order for similar reasons to you.
TrustFire R5-A3 3-Mode Memory LED Flashlight (1*AA) will take 14500's
I know the price of this one is high, but if you have not already got a lion battery charger and some batteries this is a good starter kit that comes with flash light, holster, battery and battery charger in a nice gift box and a good looking light.
Once you have a 26650 charger and battery or batteries, cheap lights in your price range that will fit it your pockets like these that can be used with 18650,s or 26650's.
UniqueFire UF-2200 Cree XM-L U2 LED 5-Mode Memory Flashlight - Black (18650/26650)
I hope this helps