I suppose the providone iodine is the common 10% solution. In order to protect your thyroid from I-131, you should take ~100..200 mg of potassium iodide per day, immediately before or during the exposure. Assuming a quarter gram per drop, that makes 40 drops of the solution for one gram of providone iodine. I'd chance a short term overexposure over an underexposure, so make that 40 drops of providone iodine per day while I hike down to the town.
On the other hand, since I was able to google all these details right now, I do have an internet connection, which means that I can contact _somebody_ and ask for a pickup along somewhere on that dirt road I would have to hike down - or, in the worst case, get the positions of the emergency workers who are active in the area and connect with them.
For one thing, the recommended dosage for potassium iodine is *really* high - something like 1000 times the usual RDA, just to make sure, as iodine is being processed rather fast. Then, providone iodine is usually only a 10% solution - and the amount of iodine in the providone is again small, on the order of 10% or so.
And my second comment was rather along the lines the knowledge necessary to get to that dosage is not something the average person would know without having to look it up - or what would be among the books I'd usually imagine being available in a cabin in the woods.