Howdy in Danmark-
I'd always assumed that since Jack still had family in Wyoming that this was the supposed reason for the trip, in the short story (SS) his mom says "He used a come home every year, even after he was married and down in Texas, and help his daddy on the ranch for a week, fix the gates and mow and all." Riverton is somewhat on the way from west Texas to Lightning Flat.
As to whether Ennis would write, his problem was he didn't have any idea where Jack was (Texas, Wyoming, Army, etc), again from the SS: "I didn't know where in the hell you was," said Ennis. "Four years. I about give up on you. I figured you was sore about that punch." Ennis didn't go far from where Jack left him in '63. Of course Ennis could have written to Jack's folks in Lighting Flat, but that would have taken the search to a different level of seriousness. Jack's card was casual, almost incidental in tone.