I suspect that various "industries" that utilize impingement cooling have developed their own specialized correlations, and it would be well worthwhile to investigate those (if applicable).
In any case, I will say as a more general statement that the shape of impingement holes does not matter directly, but could affect "h" by determining the cooling fluid velocity distribution.
Surface details of the hot body will also affect this of course, but the orientation of the hot body surface(s) are also potentially very important. I think that changing the surface for the sake of improving the cooling is unnecessary...
I believe that you could consult standard heat transfer texts to obtain some useful correlations (Rohsenhow & Choi, Holman,...), but you will have to grind through a lot of calculations if your configuration is irregular (cooling flow, surface orientation, temp. difference, etc.) in order to obtain some sort of average "h" value(s).