What happens to the put option (longer expiration) of a collar if the call gets assigned?

The put will expire and you will need to purchase a new one.

My advice is that the best thing would be to sell more calls so your delta from the short call will be similar to the delta from the equity holding.

A long stock collar (aka collared stock) consists of long stock, a short OTM call and a long OTM put. Typically, both options are OTM though there's no rule that says that they must be. Conceptually, it consists of a covered call and a long protective put. If one of the options expires at a later date then the position would be considered a diagonalized long stock collar.

If the short call is assigned early, you are left with a long put. If you started with a diagonalized collar (your example) then regardless of when you are assigned on the covered call component (early or on the short call's expiration date), you are left with a long put with a later expiration.