I don't think direct arm work of any nature is really the best choice in most instances.
http://www.youtube.com/watch?v=HKrlqsZ9qYs#t=2m30s
Now performing primary and secondary assistance movements to address certain ranges of the lift that are particular muscle group dominant, such as pause or pin presses for extra TUT in the weak range and developing starting strength, or board presses and band work to focus on the tricep dominant range of motion would be the better choice in my opinion, as these fall within the general-specific category of movements which will always have more carry-over to your comp. lifts than tricep pushdowns or LTEs. They don't even really fall in the supplemental category in my opinion. Different grip bench, floor press, CGFP/BP, shoulder presses, push presses, dumbell bench, incline and press, and incline benching with different grips would be the better choices for supplemental work too. Selecting 2-3 to rotate which address deficiencies in muscular development for you.
Generally speaking, the bench responds well to high volume and we're all capable of increasing our work capacity and recovery too. Though if you never stress these systems, how do you expect them to adapt?