Shallow well with two pressure tanks
You have the idea of it, I think, so long as you're not going for something cockamaimie like tying the air charge sides together. I don't actually think you are saying that; in any case, don't do that. 

You just set those both 2 psi below the cut-in (ie, 18 PSI for a 20/40, 28 PSI for a 30/50 38 PSI for a 40/60 pressure switch setting, with the water side empty) and connect the water line to both tanks. 

If the old vertical tank is still in usable condition, you could tap it in as well, though logic would call for a 1/4 turn ball valve; that might be a good idea for the new tank as well, since eventually it will be old, and may need service/replacement - price of the valve .vs. being able to service/replace the tank without shutting the whole system down - your call.