Need help with SNF transfer across state lines. Where do I start?

If he moves then he is allowed to change plans before the next Medicare enrollment period. Start by contacting a Medicare agent in your area. Look for local ads in places like church bulletins.
You do need to find a SNF in the area you want to move him. Since you do not mention Medicaid, I assume he is still self pay. If he is already on Medicaid then there will be many more hurtles since each state pays a portion of care. It is in your best interest to find a SNF that accepts Medicaid beds when needed.
If all boxes are checked, the transportation is next. It will depend upon in transit care needs. The costs will be on him.
Now this question is important...Can he make is own decisions? If not are you his guardian or POA

Ok so exactly what is his designation for this SNF, like is he a patient? or a resident? And how is he paying for his current stay in TX SNF?
To me, the answers are really what determines the path needed to have any “continuity of care”.

So for his current TX SNF,
1. is he custodial care private pay resident for his room and board? and what is his health insurance?. If it is a Medicare Advantage Plan, that is way waaay unusual as Advantage Plan have a defined in-network at fixed hospitals/ clinics / physicians for best coverage and SNF usually are not included in the network as their staff not in-network providers plus he can’t easily get up and go over to a in-network out patient clinic or hospital for care. So being in a SNF tends to mean care done at the NH which is not in the “network”. Which in turn becomes higher copays.
If he’s private pay, then it’s on him to pay for all costs at a new NH, so you get a fresh assessment on him done at his current NH and then look to find one in MO who can do his level of care based on that assessment and pay the deposit to do a bed hold and month or 2 of private pay; & you set up ambulance service (it too will be private pay) for the drive up.
OR
2. is he in the SNF as Custodial care resident but it’s paid by LTC Medicaid program for his room & board and then for his health insurance it’s under a MCO aka Managed Care Organization?? The MCO approach is how TX does this for Nh residents who are “duals” (so on Medicare and Medicaid for health insurance) unless it’s more rural. MCO is done similar to a Medicare Advantage Plan as it’s to is set up to be “in network” for providers. It will be affiliated with a Health Science Center or a large hospital / physician group nearby, with a NP or PA who does most of the bedsides for NH residents. MCOs are mainly run by 3 big companies in TX. Molina Health is the big player in this as they do MCO type of health care billing across the US for duals in facilities. If he’s LTC Medicaid, he will have to do a total new application for however MO does eligibility for BOTH Medicaid as health insurance and for LTC Medicaid program. That 30 Days you mentioned is - I bet - what MO has as residency requirement but IF he needs to do things to clearly establish himself as a resident of MO (new banking, new drivers license, all things to create a valid new legal residencal address in MO) that could take longer than 30 days to accomplish. If he owns any real property in TX, like home / land / car, those will be nonexempt assets for MO Medicaid. I bet the 30 days is also used for the time period you have to register any vehicles without running into penalties, as State s really want cars registered and insured in State.

FWIW Any move allows him to break out of current insurance to go onto new one in MO outside of open enrollment period without penalty.
OR
-3 he’s not custodial care but actually is a post hospitalization rehabilitation patient??? His would mean his bills can be sent to Medicare or Medicare Advantage Plan as rehab is a benefit. Rehab is time limited & tends to be 20/21 days for most as that’s the max period of time Medicare pays rehab at 100%. But can go to 100 days at 50%. What happens is they stop “progressing” so Medicare stops paying and then either they go back home or if that’s not feasible sequeway from rehab patient to LTC resident.

If he’s rehab, this will be quite challenging imho.

So what is he??? as it will be very VERY different paths to get someone moved to another State and into a NH.

Medicare doesn’t pay for SNF. That requires either private pay, a long term care policy or Medicaid. Some states have waivers.
I would go to the SNF you want to move him to and ask for assistance.

You can find the different Medicare advantage plans online.
Medicare,gov

https://www.medicare.gov/plan-compare/#/?year=2024&lang=en

Once you enter the zip he is moving to on the compare link, choices will appear. Call a few and see if they can help answer your questions on applying.