OK, I'm taking a crack at the outer perimeter of Kentucky. Source for the routes is Google Maps. Two rules of note: I did not include West Virginia "county" routes, nor did I include Missouri secondary lettered routes. I also did not include Virginia secondary routes. However, I did include Tennessee's secondary routes, signed inside triangles, because they were part of the single state system until routes were divided into two classes in the 1980s.
As before, I will note segments I have completed in their entirety in BOLD. An entry in RED indicates I have driven part of it.
Starting as I did for Kentucky, in the northeast, but at the intersection of US 23 and US 52 in Portsmouth, Ohio:
US 52
Business US 52 (Ironton)
US 52
Ohio/West Virginia state line
US 52
US 60
WV 75
US 52
The US 52/US 119 concurrency, part of ARC Corridor G, crosses the state line briefly into Kentucky twice. While the state line is marked, the route is maintained by West Virginia. It is possible to stay in West Virginia by using the old routing, which is now a county route, but I opted to stay on the main route since it is not maintained by Kentucky.
WV 49
WV 65
US 52
WV 80
At this point it should be noted that we're at Iaeger, well inside West Virginia. This was unavoidable.
WV 83
West Virginia/Virginia state line
VA 83
VA 63
VA 83
Business US 23 (Pound)
US 23
Business US 23 (Esserville/Norton/Appalachia/Big Stone Gap)
Alternate US 58
US 58
Virginia/Tennessee state line
US 25E
TN 63
US 25W
We have finally made it back in proximity to the state line, at Jellico
TN 297
TN 456
US 27
TN 297
TN 154
US 127
Here, we run into the same problem we had tracing Kentucky from the inside. The US 127/TN 111 intersection is actually in Kentucky. So this requires a bit of a detour to remain out of the Bluegrass State.
TN 295
TN 111
TN 52
TN 151
TN 56
TN 52
TN 259
US 31W
The TN 259/US 31W intersection is right at the state line, but appears to be entirely in Tennessee, so it counts.
And there is another issue here. Technically, the route closest to the state line is I-65, not US 31W, but until the new interchange between I-65 and TN 109 is opened, the route has to be US 31W.
TN 52
TN 49
TN 25
TN 151
US 431
US 41
TN 76
TN 238
TN 237
I-24
TN 48
TN 236
US 41A (as it is signed in Tennessee)
TN 374
US 79
TN 140
TN 69
TN 140
TN 54
TN 190
TN 118
TN 89
TN 22
US 45E
US 51
TN 22
At Tiptonville, there is a loop of TN 78, TN 212 and TN 22 that goes closer to the state line than the route listed here, but it requires some backtracking so I'm not using it.
TN 78
TN 79
TN 181
I-155
Tennessee/Missouri state line
MO 84
I-55
US 62
I-55
MO 80
MO 77
US 60
Missouri/Illinois state line
US 51
IL 37
IL 169
US 45
IL 145
IL 146
IL 1
IL 141
(Is this bridge still open? I know the one at New Harmony is closed.)
Illinois/Indiana state line
IN 62
US 41
I-69
IN 662
IN 66
IN 62
IN 135
IN 11
IN 211
IN 111
I-64
I-265
IN 62
IN 56
IN 156
IN 56
US 50
There's no second access from US 50 to I-275 so for a short period, I-275 is closer to the state line than US 50 but the interstate can't be used for this brief segment.
Indiana/Ohio state line
US 52 (accessed by the maze of ramps at the north end of the Brent Spence Bridge).
OH 73
US 23
And we're back at the original destination.