A technique is described herein for placing a virtual object within any type of modified-reality environment. The technique involves receiving the user's specification of plural values in plural stages. The plural values collectively define an object display state. The technique places the virtual object in the modified-reality environment in accordance with the object display state. Overall, the technique allows the user to place the virtual object in the modified-reality environment with high precision and low ambiguity by virtue of its guided piecemeal specification of the object display state.