Add the data (still with unknown/undefined coordinate system) to ArcMap. Add some base data that you know has a good coordinate system. Set the data frame's coordinate system to various "candidate" coordinate systems, like Texas state systems, state plane, etc. The base data will be projected to the new coordinate system. If the base data overlays the "unknown" data, you've found the coordinate system.
Neither of those works, by the way, but I think it might be what we call BLM zone 12N (US feet). It could be based on NAD27 or NAD83.
Projected coordinate systems, UTM, BLM (US Feet)
Melita