Check the San Antonio Sunday paper online for hunting leases. I personally have not seen a 300 acre hunting lease for 600.00 per year in a long time. Not that they do not exist, but are probably rare and surrounded by high fence game ranches or some other drama. If it sounds to good to be true it probably is.
I work around oil fields and construction sites near San Antonio and live here as well. You should be able to find a lease in west or south Texas for 1000.00 - 5000.00 a year. Price will usually dictate the quality of the lease and the quality of the landowner.
Hill country property which would be west or north of San Antonio is expensive. I think land in south Texas is priced fairly and maybe your best bet.
You can book day hunts or weekend hunts for hogs year round and I have seen them advertised for 200-500 dollars. The same with Dove season. Deer is a little different and the best deals for a weekend are usually a cull hunt where you can kill a couple does or a cull buck for under 1000.00. Typically this includes meals and lodging for the weekend, some outfits will let you kill a hog or two for free as well.
There are two gunshops in San Antonio that have websites, look their sites up and call them for more information on hunting leases. They are Dury's Guns and Nagel's Guns. These folks always have hunts and leases posted in their stores or on their websites. Ranch managers make it a point to visit these folks.
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Good Luck