The likelihood of this being approved is very low, and understanding why helps set realistic expectations before investing time in the request.
Furloughs are already rare in most state systems. When they are granted, they come with strict conditions including a verified address where the inmate can be reached and monitored at all times, typically a permanent residence rather than a temporary one. A hotel or bed and breakfast in another state does not meet the stability and verifiability requirements that furlough conditions demand.
Interstate travel adds another layer of complexity. Moving an inmate across state lines, even temporarily and voluntarily, involves coordination between two states' correctional systems and raises jurisdiction questions that most wardens are not willing to navigate for a discretionary furlough request. Permission from the warden is required at minimum, and that permission is highly unlikely to be granted for travel to a temporary lodging in another state.
The warden's office is still the right place to direct a formal request if you want to try. Write a respectful, detailed letter explaining the circumstances, who the family members are, where the stay would be, and how supervision conditions could be satisfied. Present it in the most compelling and organized way possible and let the warden make the determination.
Realistically, the better path is for the Texas family to make the trip to Mississippi to visit rather than requesting an out-of-state furlough for the inmate. In-person visitation at the facility accomplishes the connection without the jurisdictional and logistical obstacles that make this type of furlough request a long shot.