Yes it will work. However you need to plan it well and exercise some caution.
I would absolutely get trailer brakes. They have saved me a couple times. Your S10 is going to be light in the rear and difficult to control under braking without the trailer brakes.
Your S10 auto trans should have an overdrive off button or control, so that you can tow in direct or 3rd. Use this when towing. Dont even think about using od.
If you accelerate slowly and don't dog it, it should work. I tow with my 97 F150 4x4, with a 4.6 auto. It has big tires and a lift, and will not hold a steady speed in od on even level ground, so I use od off to tow. Towed 2400 miles this way to pick up a tractor.
A trans cooler is also not a bad idea. I am used to flat areas, like coastal SC, and hills make it a lot harder.