Do we want to expand this to include state highways? The Krueger Brothers have an excellent song titled "Up 18 North" (well, actually, a whole album, but also a song) named after NC 18, which goes through Shelby, Morganton, Lenoir, and their main stomping ground, North Wilkesboro (home of Merlefest).

Similarly, Ricky Skaggs has "Highway 40 Blues" [lyrics] which is named for KY 40 rather than I-40, which runs through Nashville.

Ernest Martin has a song titled "Heaven's Highway 66" which is named for US 66 (I assume) but is one of your standard songs about how there's a highway to Heaven and how sin can lead you astray, etc.

John Denver's famous Take Me Home, Country Roads is named for Maryland 117.

Tom T. Hall's famous "A Week In Country Jail" is a (semi) true story about being arrested for speeding through a small town. He doesn't mention a US route or specify a road at all, but he does say it was "a small town in Kentucky." If we can find which, that could help.

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Now, for songs which are more in the vein of what you're asking for:

The famous and oft-played fiddle instrumental Lee Highway Blues is named for the US Lee Highway system. Though first recorded by Grayson and Whittier as "Going Down That Lee Highway" most people use the shorter name. The present route covers over a dozen current US Highways (link)

Charley Cline has a song titled "Highway Headed North" about an unnamed highway from Dixie to Cincinnati, but he doesn't say which one. More than likely it's current I-75, but he doesn't say which -- but it is comprised of several former US Highways.

Wynn Osborne has an instrumental song titled "Highway 31" which is probably named for US 31, which runs through Nashville, Wynn's birthplace, and a place he likely visited often, being the son of mandolin prodigy Bobby Osborne.

Robin and Linda Williams have a whole album titled Dixie Highway Sign; here is as the titular song. The whole thing is a great album, and that song in particular is a gem. It is, of course, named after the Dixie Highway (see that link for more songs about the Dixie Highway).

A second Ricky Skaggs song is "Crossville", which is an instrumental he came up with while his bus was passing through Crossville, TN on his way home to Nashville. Sadly, he doesn't say which highway. It was probably I-40, but US 70 isn't out of the question.

Millions of corny trucker songs name-drop dozens of highways along the way. The only one which I can think of off the top of my head which refers to a specific road is Dick Curless's classic Tombstone Every Mile, which refers to the often icy and dangerous stretch of US 2A

I can provide lyrics to any of the songs I've listed above if you'd like. I'll also keep an ear out for other road songs, these are just the ones that came to mind.