Remember that VATSIM controllers work in a top-down method meaning just because a charted ATC position isn’t online doesn’t necessarily mean the airport or airspace in question is uncontrolled. An overlying controller for that listed ATC position on the chart (such as an overlying approach or center controller) likely still provides the local ATC services for the airport and airspace in question.
In your example, the Austin approach/departure controller would be the local ATC at Austin-Bergstrom International Airport (KAUS). They would be ATC for clearance, ground and tower all through one controller. Once you left the Austin approach controllers airspace then you would transfer to Unicom/122.80, since there is no center controller.
If you were flying from an uncontrolled airport to another uncontrolled airport in this airspace, then you would not have to contact ATC. As long as you don’t fly through any controlled airspace.
I recommend in this case better use Sectional charts not this ‘map’ where no other important infos are visible and also is not clear what this section is. I think that in this case (USA) skyvector.com is ok to you to see at what place as Airspace you are and what you need also call as ATC. In this case you need call lowest ATC available at your place/vicinity/airspace, generally in the air look for TWR and higher, in case of at parking for FPL clearance DEL (Delivery) and higher. You need to know what ATC handle what Airspace/service where you are actually. Be careful differ 2 different things - Uncontrolled airport type and uncontrolled airspace actually by ATC.
but this is clear from vPilot Client also, list of actual Online ATC positions is there, also depended to distance from user actual position and ATC higher authority/priority level. I feel that OP asks to be not familiar with procedure in case of ATC services and with first contact depended to his actual position.