Instead of limitating the use of this tool to one line, you should be able to make a paragraph based on the shape you use the curvilinear tool on.

For instance, if i want to write on the edge of a wheel divided in different parts, so that the text follows the shape of the circle. Let's say that the name of the wheel's part is a three lines title, when i press enter, indesign would create another circle on wich the text would follow the curve. see atached picture