Its hard to answer concisely, since there would be great local variations, depending on where in the world your protagonists bunker is located, if the country was a direct belingerent in the war and many other factors.
In the scenario where the first six month are spent inside a bunker, you can at least assume that the most volatile and thus most radiactive isotopes (e.g. Iodine 131, see here for example https://en.wikipedia.org/wiki/Chernobyl_disaster#/media/File:AirDoseChernobylVector.svg) have decayed sufficiently to the point where they no longer contribute the majority of radiation.
Most radioactive particles would have settled to the ground/have been washed out by rain after six month. This suggests a scenario where contamination in the air is minor compared to radiation from topsoil and/or incorporation by food/water.
While groundbursts are considered very dirty, the majority will be deposited within a few hundred miles in the downwind direction. While such plumes can contaminate significant areas (e.g. https://en.wikipedia.org/wiki/Nuclear_weapons_testing#/media/File:Bravo_fallout2.png), there would still remain some areas not directly affected, except in the most densely populated/hardest hit areas.
The chernobyl desaster was much worse than a single bomb test in terms of radioactivity released, still the death zone is relatively small, and the area where plants died completely was even smaller. Note that the affected areas are spread very unevenly: https://en.wikipedia.org/wiki/Chernobyl_disaster#/media/File:Chernobyl_radiation_map_1996.svg
Assuming an arsenal of ~2000 warheads, which are presumably mostly in the 10-100kt range, there should be plenty of zones left that are only affected by the fraction of fallout that has spread globally (or barely at all on the southern hemisphere).
Lastly, edible food can be grown in chernobyls worst affected zones. The main problem is to select crops that do not readily take in the radioisotopes that contaminate the topsoil. And of course even carefully selected crops have to be constantly tested - its has been done on research scale.
So in summary, there would be areas where crops would be "safe" (as in acceptably low contamination) for consumption. Its another question if your protagonist would be able to find and identify such areas though. Plants may appear perfectly healthy and may still be contaminated enough to kill you within a few month or less. A common geiger counter would probably be not sensitive enough towards the lower end to identify "safe" food. Something a little more sensitive would be needed.