  We give a simple proof of a recent result in [1] by Caffarelli, Soria-Carro,
and Stinga about the $C^{1,\alpha}$ regularity of weak solutions to
transmission problems with $C^{1,\alpha}$ interfaces. Our proof does not use
the mean value property or the maximum principle, and also works for more
general elliptic systems. Some extensions to $C^{1,\text{Dini}}$ interfaces and
to domains with multiple sub-domains are also discussed.
