The portion of Ohio law that applies to transporting firearms in a motor vehicle is ORC 2923.16
- but it boils down to this for handguns:
If you have a concealed handgun license from Ohio or from a state that Ohio recognizes, you can have a loaded handgun in the car with you as long as it isn't accessible to any unauthorized persons.
Otherwise, it must be unloaded and gun and ammo/magazines/speed loaders stored separately. Separately doesn't mean separate compartments of the motor vehicle, though, just that they are completely enclosed in separate containers or separate compartments within the same container as long as they completely enclose the gun and ammo/magazines/speed loaders separately. This is from the link above:
(K)(5)(a) "Unloaded" means ... that no ammunition is in the firearm in question, no magazine or speed loader containing ammunition is inserted into the firearm in question , and one of the following applies:
(i) There is no ammunition in a magazine or speed loader that is in the vehicle in question and that may be used with the firearm in question.
(ii) Any magazine or speed loader that contains ammunition and that may be used with the firearm in question is stored in a compartment within the vehicle in question that cannot be accessed without leaving the vehicle or is stored in a container that provides complete and separate enclosure.
And then further,
a "container that provides complete and separate enclosure" includes, but is not limited to, any of the following:
(i) A package, box, or case with multiple compartments, as long as the loaded magazine or speed loader and the firearm in question either are in separate compartments within the package, box, or case, or, if they are in the same compartment, the magazine or speed loader is contained within a separate enclosure in that compartment that does not contain the firearm and that closes using a snap, button, buckle, zipper, hook and loop closing mechanism, or other fastener that must be opened to access the contents or the firearm is contained within a separate enclosure of that nature in that compartment that does not contain the magazine or speed loader;
(ii) A pocket or other enclosure on the person of the person in question that closes using a snap, button, buckle, zipper, hook and loop closing mechanism, or other fastener that must be opened to access the contents.
So, you could have the gun in a range bag in a compartment that closes with a fastener of some kind and the ammo/magazines/speed loaders in a separate compartment within the range bag as long as it closes with a fastener. Once you've done that, the range bag can be anywhere in the vehicle you like (though probably best if it is in the hatchback, it is not required).
Hope this helps!