[Feel free to delete this! I’m not a coach. And it’s not 5/3/1. ]
It’s not a template, and it’s not endorsed by anyone else, but I have found the following an effective, simple way to add more deadlift volume with low opportunity cost — just swap in light trap bar deadlifts (the raised handles) or light rack pulls for part of your “general” warmup. The bar start position will be high enough that with light weight there’s not any serious flexion in your (at least my) joints, so there seems minimal risk of problems. However, it’s a real movement, so (1) I am definitely warmed up at the end and (2) the extra sets seem useful in terms of laying a foundation.
I do it on every lower body day — a few 15 rep sets, strict, not pushing of 135, a few 10 rep sets, strict, not pushing of 225, both super-setted with hanging leg raises, possibly bodyweight bench squats. I usually alternate low-trap, high-trap, rack, cambered squat bar deadlifts for variety each workout.
I’m definitely ready to go at the end. And as a nice side-effect, my grip strength has increased, even though the weights are low.
As a mental benefit, I find “tiny” warmup stuff really boring, so warmup always feels long no matter what I’m doing. Light trap deads warm me up just as well, but feel useful, so the time goes by quickly. You get a lot more deadlift volume in, without much additional time or headspace.
It’s made a real difference, in a short time. Perhaps you can view it as a very lightweight version of BBB for lazy people.
Anyway, I’m not in a position to coach anyone, so use your best judgement.