I don't know of any law in OH that would forbid a legal, licensed permit holder from carrying their psitol as well as a backup gun. And he wouldn't have to keep it in the trunk...on his person, in the glovebox, in the center console would probably suffice but different states do have differences regarding that - I usually reference www.handgunlaw.us
The question came up in an offhand discussion several years ago, . . . some of us were discussing different strategies for SD / HD and multiple weapons hit the stage as a topic.
One voice quoted the back of the actual CHL license Ohio issues, that states: "....... is permitted to carry a concealed handgun in the state of Ohio.....".
We discussed whether "to carry a....." was exclusive of multiple carry as it seemed to define only one avenue, . . . that of carrying one and only one handgun on one's person at a time.
That was the question I posed to the sergeant who was at that time THE responsible person for issuing CHL's in Marion County, Ohio. Sheriff Baily has a reputation for not having un-knowledgeable persons on his staff, . . . so I took it that I was being told the correct answer.
Additionally, . . . most of Ohio is reasonably sane, safe, and a good place to visit, live, etc. HOWEVER, . . . Cincinnati, Cleveland, Toledo, and especially Columbus are distinctly NOT gun friendly. The mayor of Colubmus, especially, is a dried up little maggot of liberality, and pushes downward through his "chain of command" any and all efforts to curtail any firearm related item.
They would treat you as a terrorist at any opportunity, . . . so I advise all my friends and acquaintances to err if need be, but err on the side that has less consequences. In this case, . . . multiple gun carry has been pointed out by a knowledgeable LEO to be verboten, . . . might be good to go that way.
May God bless,