if you have the option ,its generally rather helpful to learn to shoot SKEET, as it helps you gain the rapid hand/eye coordination required to use a shoulder fired weapon on moving targets, now obviously use of a shotgun on Arial clay birds is not the same as shooting at a rapidly departing white tail deer's heart/lung area, but if you can consistently hit 20 plus skeet targets with a scatter gun, you'll have very little difficulty using an accurate rifle to drop a running deer at under 50 yards or so.
and used with a sling, if you take a second or two longer, with a decent optic (low power scope)
(hitting a 6" orange dot target at 100 yards in under 5 seconds,
with a first or second shot is a skill that's fairly easy to acquire with enough repeated practice)
now a lot of guys might suggest you select a lever action, pump or semi auto, so you have nearly instant repeat shot capability,(which certainly might be helpful)
but its that first well placed shot that's much more critical to upping your odds of success, than laying in a hail of projectiles on a bouncing white tail deer,
as it rapidly seeks to put a great deal of distance between you and his current location.
BTW, a good bolt action, with a fast 60-72 degree bolt lift, like a, tika, browning or Weatherby can with practice allow a well practiced rifleman to put down a very rapid follow up shot should it be required , and yeah, if you select a lever action, pump or semi auto, you can get off a second shot rapidly, but again the goal is a well placed first shot, and a 150-165 grain bullet in your 308 win is certainly a lethal combo in your current rifle IF you place it correctly.
theres no need for you to change weapons, but I'd strongly suggest you practice shooting off hand rapidly from a standing position.
the fact that your deer was not mortally wounded and did not drop within 20-50 yards of bullet impact is almost certainly the result of less than ideal shot placement (it happens, ) the lesson learned should be... your not just trying to shoot faster,
your trying to be certain that any shot you fire hits exactly where you intended it to, and
that requires practice in shooting from standing field positions, and a good sling and a decent scope certainly aids in precise shot placement!
BTW
your certainly not alone, anyone with much experience has either missed a shot or had it hit in a place it was not intended
and I can tell you most of the guys Ive hunted with can shoot 1"-2" 100 yard bench rested groups easily, but damn few could could consistently and rapidly hit a 5"-6" orange dot at 100 yards shooting off hand until they put in a great deal of practice, and everyone of them became much more consistently successful once they acquired that skill level,
and yeah, if you have the option of leaning against a tree or dropping to a sitting position obviously its to your advantage to do so, so you do that!
https://tpwd.texas.gov/education/hunter-education/online-course/images-shooting/sittingRao_20120413_0017.png
https://tpwd.texas.gov/education/hunter-education/online-course/images-shooting/StandingLee.png
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