I am not talking about the Oleśnica near Wrocław. I am talking about a small town of the same name which is Oleśnica (Staszow, Swiętokrzyskie). I am wondering if there is a bus or some form of transportation from that small town to someplace larger (like a city that is served by a train).
Yes, there are buses to Busko-Zdroj (which I think has a station with trains to Kielce and perhaps Krakow). Busko certainly has fast buses to Kielce.
And even a bus to Kielce itself, if you don't mind getting up early.
e-podroznik.pl is an annoying website, but once you've figured it out, it can be useful.