Just adding to the great responses above - what you're looking for is probably better referred to as a "coworking space" vs. a virtual office if you are googling for a solution. Virtual offices tend to be more focused on mail service although some are bundling day passes and meeting room time.
Regus is your best bet if you really need broad national coverage. They have more locations than 7-11 :-) But they are very "corporate" in design. There are coworking spaces in every city and most offer day passes for non-members. They just take more work to find. At this point there is no real "network" of them.
I also advocate using liquidspace. They have a fantastic app and you can search by type of space. They do not own the spaces - they are a space finding service.
Full disclosure - I am a coworking space owner with locations in Chicago and Palo Alto.
Congrats on your business expansion and good luck finding space!