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What is your objection to contacting the recruiter? As already noted, a new confirmation will be sent when the itinerary is changed, so the company will find out whether you ask in advance or not.
Before you try anything, look at things from the other side of the arrangement. Would you be eager to hire someone who went behind your back to obtain a personal benefit from something you were providing to the prospective employee? Would you wonder whether the candidate was serious about the job or just using the interview as an excuse for a nearly-free vacation?
I'd strongly advise calling the company and explaining what you want to do. Presumably if you need to fly there for an interview you will need to relocate if you accept the position. It is perfectly reasonable that you'd want some time to look around a city that could become your new home. Unless this is a multiple-interview process where the looking around is supposed to wait until a later step, the company should be willing to make the change. They might even pick up the room cost, if you aren't staying with one of your acquaintances.
In any event I find it disturbing that this particular question was asked and I hope you can see why.
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