The warden is not necessarily your first or best move, and calling repeatedly from the outside is unlikely to get you far. Facilities are not set up to field that kind of inquiry from visitors, and it tends to go nowhere fast.
The more effective path starts on the inside. Have your inmate go directly to the chaplain. The chaplain is typically the person who handles marriage requests at the facility level, and they can tell you quickly whether the inmate is even eligible. Not everyone is. Disciplinary status, sentence length, housing classification, and facility policy all factor in. Finding that out first saves you a lot of wasted effort.
If the chaplain confirms eligibility, the process usually involves a formal request that works its way up through administration. That is where the warden or facility administrator may eventually sign off, but it typically comes at the end of a process rather than as the starting point.
Persistence matters, but it needs to be directed at the right people. Your inmate pushing from the inside through the chaplain will move things faster than phone calls from the outside to a front desk that has no authority to help you.