The best way is to simply get out there, the more you cover the more likely you are to find a new cool place. Although you say there isn’t any public near you, and texas is especially bad for that, some researching will find you spots.
First off, don’t worry too much about puling over on dirt(or paved) roads, many of us naturalist do this all the time. While it is fully legal to use the shoulder of a road, just make sure to be safely away from traffic and that you won’t get stuck trying to get back on the road. If I am going to take more than an hour I will sometimes leave a note in my windshield explaining what I am doing. If worried about safety/visibility or how you will be perceived consider wearing a high-vis vest/shirt.
One method is to go where others go, you can use alltrails and strava to find walking paths that might not be obvious. Your state might have a trail maintenance organization that publishes maps of trails (for example: https://www.wta.org/go-outside/map). Caltopo and open street maps often have walking paths marked, just make sure to check if the land is publicly accessible or get landowner permission. You can check who owns the land (in the usa) through your county property assesor/appraiser website which should have gis map property search feature.
You can find habitat maps/natural area inventories from state and national agencies, if there are unique/rare/endangered habitats try to see if you can access them, either on public land, roadside, or by talking with landowners. You can look at localities of herbaria collections, your state or region should have a herbarium with collections from the area, and most herbaria have websites that let you search specimens. Your county should have a parks webpage, and so should your state, make use of them.
Make other naturalist friends(or message/email), and gain knowledge of spots from them.
If you have access to a boat/kayak or other floating vehicle, most waterways in most states are public property (look into local laws) so exploring riverbanks can be a good way to get access.
Lastly if you are willing to break laws and trespass that opens up a whole new world to explore. But I won’t be giving tips for that on this public forum :)…