There are quite a few book of actual logging songs, e.g.,

Earl Clifton Beck, Songs of the Michigan Lumberjacks
E. C. (Earl Clifton) Beck, Lore of the Lumber Camps
William Main Doerflinger, Songs of the Sailor and Lumberman
Edith Fowke, Lumbering Songs from the Northern Woods
Roland Palmer Gray, Songs and Ballads of the Maine Lumberjacks
Franz Rickaby (edited by Gretchen Dykstra and James P. Leary), Pinery Boys: Songs and Songcatching in the Lumberjack Era

The Traditional Ballad Index currently has 204 songs with the keyword "logger." The most popular are, in descending order:
The Jam on Gerry's Rock [Laws C1]
Peter Amberley [Laws C27]
The Logger's Alphabet
Canaday-I-O, Michigan-I-O, Colley's Run I-O [Laws C17]
The Shantyman's Life

"The Jam on Gerry's Rock" is by far the most popular -- I have 67 citations, to 35 for "Peter Amberley." Steve Roud has 220 for "The Jam," 114 for "Peter Amberley."

I highly recommend the tune sung by A. C. Hannah and collected by Franz Rickaby. One of the best tunes I know, for anything. A recording is here: https://mnheritagesongbook.net/the-songs/texts-and-additional-notes-on-printed-songs/the-jam-on-gerrys-rocks/

Obviously if you want happy stuff, the Sailor's Alphabet would suit better. :-)

If there is a particular forest-y topic you are interested in, of course, we can try for something more specifix.