My understanding is that both would need to keep the Solu-Cortef on hand. 

The main difference in treating the two forms day-to-day is that people with the Salt Wasting form must take Florinef in addition to the Cortef, to help them retain the proper salt and fluid balance.  Although I believe that I’ve heard of some non-Salt Wasters benefitting from Florinef as well.

In an emergency with adrenal crisis though, the person would need Solu-Cortef, regardless of the type of CAH they have.