All prior work on measuring the spins of stellar-mass black holes (BHs) via the X-ray continuum-fitting (CF) method has relied on the use of weakly Comptonized spectra obtained in the thermal dominant (TD) state. Using a self-consistent Comptonization model, we show that one can analyze spectra that exhibit strong power-law components and obtain values of the inner disk radius, and hence spin, that are consistent with those obtained in the TD state. Specifically, we analyze many RXTE spectra of two BH transients, H1743−322 and XTE J1550−564, and we demonstrate that the radius of the inner edge of the accretion disk remains constant to within a few percent as the strength of the Comptonized component increases by an order of magnitude, i.e., as the fraction of the thermal seed photons that are scattered approaches 25%. We conclude that the CF method can be applied to a much wider body of data than previously thought possible, and to sources that have never been observed to enter the TD state (e.g., Cyg X-1).