Like all centum languages, Greek lost the contrast between sequences of a velar + *w and labiovelars, so you're not going to find instances of a velar followed by a digamma: in all Greek dialects of the alphabetic era, those would have become π/β/φ, τ/δ/θ, or κ/γ/χ depending on the dialect and context.
The reason you only remember words beginning with w is because those are just the most prominent on account of their obvious effects on metre in (primarily) Homer. TKR gave some examples of words with intervocalic digammas, but one famous word with a non-intervocalic word-internal w that we can be quite certain of is Attic κόρη, Ionic κούρη < Proto-Greek *kórwā, as well as the male equivalent Attic κόρος, Ionic κούρος < *kórwos (the difference between the dialects here is entirely due to the different way that w was lost), which are attested in Mycenaean as 𐀒𐀷 ko-wa and 𐀒𐀺 ko-wo respectively. (I'm not aware of any direct attestation of the digamma there in alphabetic Greek, though.)