How can I keep the shape of my free-form bread dough when it is rising?

Jay's answer of a couche is great for relatively long loaves like baguettes or even ciabatta. For oval or round loaves, however, you'll need support on more than two sides. In that case, the solution is a basket known as a banneton or brotform (depending on which language you prefer to make your bread in). Again, they are not for baking -- you dust them with flour and then gently flip them to get the dough out right before baking. (Note that this is how bakers often make those lovely patterns of rings of flour on the top of their loaves.)

If you don't use either, you may still be able to achieve greater height by shaping more fully. Many people have been cautioned by various books to shape very gently and avoid degassing. I spent years producing flabby flat loaves this way, thinking that the key to loaf height was avoiding degassing and letting the bubbles grow. But it's often the opposite: by leaving the bubbles too big and shaping gently, you don't stretch the gluten enough to provide support and the yeast actually grow more slowly since they are immersed in their own waste product gases. (The only time to shape gently is if the yeast isn't strong enough to raise the dough again, which can sometimes happen with long sourdough fermentations or with very rich doughs like brioche, for example.)

While it's good advice to avoid unnecessary degassing (you don't actually want to "punch down" the dough to shape it), the main way to achieve a tall loaf is by having a very taut "skin," and that generally can only happen if you shape forcefully to stretch that "skin" multiple times. When I make free-form loaves without a banneton or other support, I generally:

1. Cut the dough into pieces after bulk fermentation.
2. "Preshape" the loaves roughly into rounds or ovals by folding repeatedly to tighten the "skin." (Stop when the skin feels taut, and definitely stop if it tears.)
3. Bench rest for 10-15 minutes. (This allows the gluten to relax, before being stretched one last time.)
4. Shape very tightly, again being sure to pull the skin quite taut.

For more shaping advice, I'd recommend looking at Jeffrey Hamelman's Bread: A Baker's Book of Techniques and Recipes, which devotes a 30-page chapter to shaping techniques, with a lot of drawings to illustrate the individual steps for each particular loaf shape. (He also has another entire chapter on braiding techniques.)

Also, to increase the "strength" (elasticity) of the gluten in the dough, you could also try incorporating "stretch-and-fold" manipulations periodically during your bulk rise. This will prepare the gluten even more before shaping.

Lastly, using a baking stone or steel can help in inflating the dough fast enough in the oven before it has a chance to spread even more: you want the bubbles in the dough to blow up like a balloon quickly during baking. If they bake too slowly, the bubbles can collapse or the gluten will stretch out and allow the dough to slacken before the crust hardens.

A lot of free form bread will generally need a little support when it is rising. The tool to use in this situation is a baker's couche. Essentially it's thick cloth typically made of flax linen to support the sides of the dough. The flax linen is flexible enough to form around the dough but thick enough to hold its form.

This is one example of a baker's couche.