The problem is that you want a totally a new aircraft, the owner will want you to have some quality hours ( usually 10) in one before they will rent it to you - I suggest you take the trip in an aircraft similar to what you have been training in - i.e. if a warrior but you have a retract and csu rating then rent an Arrow. If you haven't done quite a few circuits in a cessna 182 you will find it doesnt take much effort to crunch the nose wheel down etc.
To take an aircraft on a long trip for a week or so you will need to find an organisation that has the relevant aircraft but is commercially quiet and needs the rental time. You will have to impress them with your airmanship and risk profile - if they like you it will be easy, but that usually means they know you.
Low hour PPL's are risky for the owners - you will need to demonstrate to them that you are entirely safe!. For me on my first few long trips away my own, I had to make a real effort to not cut corners, do all my weather checks and planning meticulously etc.