Deck Size Problems in Deckbuilding
The best way is obviously subjective so I'll throw out some ideas for Severance. Here's two quick ideas to make Severance possibly work during the initial reveal:

A): When you reveal your card, orientation matters. Straight-up means play as normal, sideways means severance. This is a simple way to do it but opens up the door to misplays of "Oh, meant to fire this guy but placed him straight" or "someone slow-rolled his flip to get extra information" and what not but has the upshot of no additional pieces to keep track of. 

B): Have two additional meta cards, one that says "Keep", the other "Fire" (similar to the voting tokens used by The Resistance). You would play one of these face-down along side your employee and reclaim it at round's end. This has the benefits of no physical ambiguity like option A but does require another set of components for everyone and adds another step when you intend to keep your deck intact most of the time. If you intend pruning to be an integral part of the game, maybe not the worst option.

If you don't mind having Severance happen at a later phase, this opens a few paths. Firstly, you could have the Severance effect be defined by the staff themselves. This allows them to trash additional stuff in different zones. E.g. 


Mook A has "Do thing_1. Severance {cost}"
Thug B has "Do thing_2. Severance {cost} and trash a card in hand"
Intern C has "Do thing_3. Severance {cost} and trash a card in your discard pile"
Fundraiser D has "Do thing_4. Severance {cost} and trash a card in your discard pile"


Alternatively, there's also the option of having Severance be a static ability, (e.g. only trash current played card), which allows certain staffers to have abilities that modify it. E.g.


CEO E has "Do thing_5. You may trash something from your hand/discard pile instead of CEO when severing it. Severance{cost}"


Just some thoughts