I was looking at usends.com on the US 290 page. It's always bugged me that a historic endpoint would just disappear. I've looked at the area on Google Maps a few times. Today I noticed a hill that looks like the hill in the distance on the post card, which has a somewhat distinctive shape. Matching the positioning of the hills in the photo, I believe the photo was taken from about here:
https://goo.gl/maps/y2oAMycWV2M2I wanted to find as much detail as possible, and I noticed that the topographic map shows windmill 3978 right about there, and that seems to be at the location of an old round water tank that's still there, but that tank is on a hill, so it doesn't match the shape of the ground in the photo.
https://goo.gl/maps/eEo56y5ruFnThen I remembered that TxDOT put their right of way maps online, so I decided to look for clues there. The map shows that, as I suspected, the old pavement to the west, near the westbound-to-eastbound turnaroud is the US 80 right of way.

It also shows that the scarring on the ground toward the southeast across the interchange was US 290.

As can be seen on the map, the new I-20 ramp somewhat follows the old US 80 right of way, and the old junction shown in the photo is just about where that scarring meets the modern road. It looks like it was a little to the north of there, where the ground would have been reshaped to build the new road. Conveniently marking the spot, it's right where the road narrows to one lane.
The map even shows the details of the buildings. It shows the service station at the junction, with the water tank tower behind it. It also shows a motel to the east of it, which can't been seen in the photo, and may not have been there at the time. There were other buildings nearby, including dwellings, presumably for the staff. That would make for an easy commute.

The ROW map is at
https://maps.dot.state.tx.us/ROW_PDF/ODA044109AA.pdf The page to get maps is
https://maps.dot.state.tx.us/AGO_Template/TxDOT_Viewer/ The maps are not available for all highways, and in some TxDOT districts very few are available.