You'll have to see what works for you. Personally I've only ever really put on muscle by working mostly in the 1-5 rep range and eating a ton.
Also, when you say you do a lot of volume, what do you mean by a lot? For example, if you're hitting three sets of 12 once per exercise per week, that's not really that high volume. It isn't low, but it isn't high. You're doing a fair bit per set, but over the week you're averaging 36 reps per exercise. If you do 5x5 and increase the times you hit certain lifts (squat, bench, DL, rows, press) to twice a week, your reps per week for each lift go up to 50. You do less per set, but much more overall.
The other question is what you spend you time doing. For example, if you do a bunch of isolation work AND compounds all in that 12 rep range, it adds up to lots of reps in total - but still not necessarily lots of reps spent on what will drive your growth the most, which generally speaking are big compound lifts - squat variations, DL variations, pressing/benching variations, rowing movements.
Your idea of going to 5x5 is definitely worth a try, simply because you'll get stronger, which will let you shift more weight for more reps. However, what I'd recommend would be instead doing something like 5/3/1 Boring But Big and eat a ton of mostly good food - do that and you will hit BIG volume on the big money lifts and it'd be a shock if you didn't grow from that combined with the increased food intake.