No, that's not true in my experience. The recipe-related things that make dough spread less (I have a whole video on the topic) are: (1) the amount of leavening; (2) how much flour is in the recipe relative to other ingredients (like fat and wet ingredients); and, to a lesser extent, (3) the type of fat used (shortening vs. butter). Cream of tartar actually acts as an activator of baking soda, so, if you use baking soda in your recipe to leaven, it also needs some cream of tartar (or another acidic ingredient) to allow the baking soda to do its leavening job. So . . . if you add cream of tartar, if anything, it may enhance leavening and spread depending on your leavening agent (powder or soda). Cream of tartar is also a stabilizer of egg foams, which is why some people use it in their royal icing.
So, I would suggest: increasing the amount of flour (or adding cornstarch) and/or reducing the amount of leavening in your recipe, but doing so gradually, as both can have a toughening effect on the dough. Also, if the fat in the dough is butter, or dominantly butter (not shortening), the dough may respond to chilling before baking. (Shortening-based doughs don't respond much at all to chilling, but they also spread slightly less than butter-based doughs, all else being the same.) Moreover, the thinner you roll any dough, the better it holds its shape.
Again, I've got a whole video that discusses all of these recipe- and process-related variables as they relate to 3-D baking of contoured shapes, but the same principles apply to 2-D cookies. You might check out my channel; it's the most recent video I posted.