I am doing the same thing, I have to use other clinics when away from Ft Worth.
You can use any VA hospital or clinic, you just have to register as a temporary patient. Depending on what you need (blood work like me, or whatever) you will get a clinic to go to. I've done it four times and never had a long wait but all I am getting is an INR test.
This is supposed to change with the new card that was sent out but I haven't tried it and I have a few doubts about it working.
Call your Hospital that you are registered at and speak with a patient advocate and get a direct answer. three of the times I did went well with no hassle until I met one of the long term disinterested clerks.
If you have any questions feel free to PM and I will try and help.
Travel Safe.Bob