Which swap curve are you trying to interpolate? And what swap are you trying to price? The 60d to 1y60d swap (1y long, starting at 60d), or now to 1y60d (not a usual length)?
Really by interpolating the swap rates, easy though it seems, you are implicitly building a curve of forward rates. You are also ignoring the structure of the market where the fixing rate (e.g. a Libor) is not the rate used for discounting (typically an OIS).
The process of constructing a curve of forward rates from market prices is called bootstrapping or curve calibration, and the 'interpolation' you describe would be pricing from that curve.
There are entire systems to provide this calculation for you, or libraries you can use to calculate the rates using a viable market model. The question is really whether you want an exact, defendable answer or an approximate one.