In many states, the true name of the hospital is typewritten on the full copy version of the amended birth certificate that any adoptee can obtain. This format has the names of the adoptive parents and at least a space in which the name of the hospital could have been entered. The addresses of the vital records offices in each state are online at :
http://www.cdc.gov/nchs/howto/w2w/w2welcom.htmIf the name of the hospital is obtained, one can search for it and ask if they have retained the old medical records. If they have, apply for copies. I know of an adoptee that received a packet of records from her hospital, including the name of her birth mother, and a xeroxing bill for $55.00!
I don't know if the local newspapers in Cheyenne were too sophisticated to bother printing the names of all babies born each day, but I have heard of it happening elsewhere. Other sources of a birth name would be baptismal records and duplicate birth records at a local registrar (if any) who may or may not have bothered sealing the original birth record. Ancestry has a message board for Laramie County :
http://boards.ancestry.com/mbexec?htx=board&r=an&p=l...Maybe the genealogists who hang out on that board know if there is a local registrar for that county and if he allows them to walk into his office and examine any records they want.
Reg Niles