These Teams come to mind.
Chillicothe D3
Logan D2
Zanesville D2
Marietta D2
Waverly D4
Williamsport Westfall D4
Wheelersburg D5
Wellston D4
Ironton D4
Portsmouth D4
Circleville D3
Crooksville D5
Glouster Trimble D6
Nelsonville-York D5
Amanda Clearcreek D4
South Point D4
Rock Hill D4
Portsmouth West D5
Washington C.H. Miami Trace D4
Chillicothe Zane Trace D4
Newark Catholic D6
What Larger Teams could use a change and play up instead of playing down concerning their non-league schedule in getting ready for league play. Also, the competition might help the teams benefit by making the playoffs.

If I would mention a few, just in our SE-Ohio area, these 5 come to mind.
1. Jackson who is in our SEOAL is one. They play Wellston and McArthur Vinton County year in and year out.
2. Gallipolis (Gallia Academy) is another in our SEOAL who has played Pomeroy Meigs, McArthur Vinton County and Point Pleasent, WV reguarly. This season they have kept The Plains Athens on their schedule and have added Wellston.
3. Vincent Warren Local from our SEOAL who plays Belpre regularly.
4. Ottway Northwest from the SOC II who plays Portsmouth East (Sciotoville Community) on a regular basis.
5. Greenfield McClain from the SCOL who plays Chillicothe Unioto and Frankfort Adena on a regular basis.