Our state’s Department of Taxation and Finance issues checks drawn on a bank account that has an out-of-state routing number (non-local). Do we have to figure out a way to give next-day availability? Can we require the use of a special deposit ticket for those checks?

How disloyal of your state to use an out-of-state bank! Of course, there must be mitigating reasons for their choice.

These checks qualify under §229.10(c) for next-day availability, regardless of where the drawee bank is located. You may condition that next-day availability upon your customer's use of a special deposit ticket.