Would flooring underlayment work as a soundproofing buffer?
 

I've already got the weatherstripping at the bottom of the doors. And since there's an office right next door, I'd like to put the self adhesive underlayment to absorb some sound between the 2 offices.

What do you guys think?

Not much, I would guess, unless it's sandwiched between an extra layer of drywall.

You could spray foam outlets, use thin area rugs, and have a thick wall hanging behind the desk, but without major work by the owner it's as difficult to stop sound as it is to hold water.
Thicker doors have the most effect on the high ranges and hallway transmission.

Or, stop banging the secratary during business hours.

Check your ventilation ducts first. There is sound deadening drywall. Sound proofing an office is multi-step. I did some child abuse interview rooms.