We'd need to know more about the PSU to say whether it's a good idea or not though. 500W is sufficient in theory, but the quality is where it really matters -1060's require additional power from the PSU, so there would also need to be available (and appropriate) PCIe connectors.
Can you provide the make/model or even serial number of the PSU?
I don't know much about Cougar PSU's, but I believe most are made by HEC. While it's not likely to be the best PSU ever made, it's likely far from the worst.
There should be a model number on it (RX, CS etc, I think?).