searching for William Klein (moms real father)

My mother Ruth Peterson (maiden name Olson) b; 6-13-1942 d; 3-5-2007 was born to Clara Peterson. Mom was then adopted by Orville Olson who married Clara in September 1942. Mom's biological father was named William Klein. Unfortunately, that is all the information I have on him. The adoption most likely would have taken place in 1942-43 and I think either in Jackson County, Wisconsin where the family lived or in Milwaukee, Wisconsin where my mom was born. I have a lot of info on the other family members if needed, just not on William Klein. My question is, does anyone know of way to figure out who exactly William might be with out spending lots of money on fees? It's like a needle in a haystack to me!!!! Thanks in advance for any help.