I don't know where my copy currently is, but I suspect that you will find that info in
Neyland: The Gridiron General by Bob Gilbert. Originally published in 1990 by Golden Coast, this book has long been out of print, but, if it is that important, you should not have any difficulty borrowing a copy via inter-library loan. Assuming that an archivist is employed by the
Knoxville News Sentinel, you could contact that person and ask him/her if they could determine through a search of 1936 issues precisely when Neyland returned from Panama.
Incidentally, is your book supposed to be biographical, historical or historical fiction? If it is historical fiction, that is the kind of detail about which you could exercise some literary license.
For the record, the 1935 game was the one in which Bear Bryant played against Tennessee with a fractured fibula.
P.S. According to this article (
News Sentinel archives to be digitized),
KNS is in the process of digitizing their collection of old issues of the paper. As of April 01, 2016, "back issues of the newspaper are only available on microfilm, organized by date. In the new system, readers will be able to search entire years from 1922 to 1990 by names and keywords. The Knox County Public Library Foundation . . . has completed a $650,000 fundraising campaign for the project thanks to a newly-formed partnership with the University of Tennessee. The relationship means UT students, faculty and staff also will have access to the digital archives along with Knox County Library card-holders. . . . The library has digitized two years so far, 1940 and 1982."