That looks like a projected Coordinate System, with units in either feet or meters.
You can figure out what coordinate system your points are in by trying out different coordinate conversions on this website.
Enter the lat/long of a coordinate, choose the target CRS, and click convert. Try out different target CRS's until the conversion gives you the known coordinates.
Given the location, start by testing CRS's with Texas in their name, eg:
- EPSG 3082, NAD83 / Texas Centric Lambert Conformal
- EPSG 3081, NAD83 / Texas State Mapping System
It shouldn't take long to test all 10 Texas-specific CRS's. If none of them work, expand the search to US-specific CRS's, and North American-specific CRS's.
Remember that the US often uses US survey feet or (international) feet for the unit of measure. When you see large numbers like that, check the foot-based coordinate reference systems first. Austin falls into the Texas Central zone in the State Plane Coordinate System, maybe try that one first. You will find it difficult to determine which geographic CRS is being used. It's probably one of the NAD 83 ones, but there have been several re-adjustments and coordinate differ at the centimeter to decimeter-level.
An alternate approach is to figure out where the origin of the CRS is. Then you can limit your search to CRS's with that origin. Find the origin by measuring from the known point, 3109020 units to the west and 10114224 to the south. Since the data is in the US, you'll have to test it with both meters and feet. If the origin is on the equator you're probably looking at a UTM projection.