I used to have a ramp set up to put my Mini in the back of my long bed F250. It fits, but going down the road at anything more than 35 mph is an experience I don't care to repeat. All the weight in the back tends to make the front end really light and it gets sketchy on the highway. Tying it down was also a challenge.

Flat towing is an option - I personally have flat towed several thousand miles, and yes on this most recent move from Saint George Utah to Atlanta around a two thousand mile trip shredded a tire after around a thousand miles.  No problems towing it from Atlanta to St. George.  Now was it the tire - too much flex in the sidewall, did it get a nail or what.  I did notice Harbor Freight has a Haul-master 5000 Lb. Capacity tow bar and with a 20% off coupon puts its price in the $60.00 range.  So with some additional parts - things like some grade 8 hardware and making the mounts to adapt it to a Mini you have say $100 - $150 in it.  I also notice an article out there on the internet - Flat tow instructions. Using grade 8 bolts/washers/nuts/spacers/etc., available at Lowe's, and some angle iron how to adapt  the tow bar to a Mini.
     I also am looking at the other options what is available in the Atlanta area for towing, I did like the look of Randy's aluminum trailer, thought about the trailer listed in the for sale section here but the 10 hour drive to look at (in California) drive back I decided against it.

I’ve only flat towed with an “A” bar. Cheaper then renting a trailer otherwise contact DrMini

I have also towed once from California to Colorado on a trailer but I don’t remember much about that trip. Then I towed or moved my two minis from Colorado to Washington in a moving truck. Just some other thoughts

You don't need an expensive aluminum car trailer. Get a 5x10 landscaping trailer with the built in ramp on the back. Perfect size for a Mini. Home Depot or Lowes or Tractor Supply Company or the ElCheapo tool stores all have them. Or find a used one on Craigslist and save a few bucks. 

Thanks for all of the feedback so far. I'm trying to balance cost and utility here. I don't envision using the trailer much once I'm back home but I think it's correct to say that you get most of your money back on an aluminum trailer. There's still a few months to decide. It may be a game time decision based on craigslist inventory. Unless anyone happens to already be running the route and wants to make some bucks.

Interesting topic! I bought an aluminum car hauler(open). Its a bit longer than is needed for a Mini, but it was on sale. Its double axle, which means its much more predictable when backing up...and its much more stable when towing down the highway....no fishtailing. 
A friend of mine used to tow his Mini back and forth from Denver to NJ using a tow dolly! Not a good idea!! On one of his trips, one of the rear wheels literally fell off of the mini!! 
I know if you go to UHaul, and you rent one of the open deck car haulers, you have to tell them that you are towing a MG Midget, because I tried telling them that I had a Austin Mini and they couldn't find it in their books!! But that would work also.
IF you are planning on using the trailer often, it may be best buying one. You can ALWAYS get your money back(or most of it) when you go to sell it...especially if its aluminum!! Just sayin...

I used a basic 14 foot wood floor flat bed open trailer. Not super heavy and pulled the trailer and mini with a Lexus RX 300 over 800 miles and you could barely tell I was pulling anything. The F-150 will be great. Just watch how you tie it down, I've seen the rear subframe of a mini crushed by tying down to it too tight.

My first mini move I rented a Uhaul car trailer which was expensive and a giant pita on a one way rental. After that I bought a 6x12 utility trailer to tow my mini, I now have a proper enclosed trailer. For a one way trip, buy the utility trailer in GA, they weigh under 1000lbs and generally have a 2000+ capacity more than enough for a mini. The only issue I ever had was opening the doors once the car was on the trailer, it can be a tight squeeze against the fixed sides of most trailers. I towed the utility trailer with a '93 V-6 Ranger, the F150 will tow it with no problem.

The Utility Trailers for rent at Uhaul are 1) going to be expensive and 2) I believe the sides are pretty tall making the strapping down of the car a significant challenge. I just helped load a mg midget onto a Uhaul utility trailer and it isn't impossible, but it is not as simple finding tiedown locations as a flat bed would be.

Consider me a +1 for buying something in GA and selling it in AZ.

Another slightly more obtuse option for a full size truck is building some wood ramp/shelves on top of the wheel wells in the bed to make a flat shelf on each side with 2x6 planks and carry it in the bed of your truck/ This will require that you have the tailgate down the whole way and also a loading dock at the point of loading and back near your home.  Of course you can build your own ramps, but loading dock will make it much easier.

Another method is to ask Dr mini on this site to move it for you, he has many satisfied customers here.

I have towed my Minis on my 10' Aluma aluminum trailer for thousands and thousands of miles. The trailer has a 2200# single axle. Three of my Minis are long-wheel variants. I even carried a 1918 Model T Touring car on that trailer from the Seattle area to Phoenix. Yes, a 12'er would give extra wiggle room. Way back when, I towed my first Mini 12000 mis. 4 wheels down behind my first GMC motorhome without a single issue. With the coach having front vs. rear wheel different track widths and the Mini a third and much narrower width, coming to something lying in the road ahead or a pothole always left me with a decision to make very quickly. I never had a tire fail on the Mini or on the trailer. A friend just recently was towing his Mini 4-wheels down and it blew a tire, so that should be considered. Randy

Hi guys, I'll be bringing home my father-in-law's 1962 Austin Mini Mk1 in the coming months. The means towing it behind my F-150 from Atlanta to Phoenix. I don't own a trailer and I'm thinking the most cost effective answer (without flat towing) is to rent a u-haul utility trailer. Failing that option, I will look at buying a trailer in Atlanta and keeping it or selling it when I get back to Phoenix. 

The question is this, what is the smallest possible trailer that reliably holds the Mini. My concern is that smaller trailers generally aren't rated for highway cruising but larger "vehicle trailers" are heavy and take up a good portion of my 7500# capacity before even loading the Mini.

I'm sure you all have a vast amount of experience in this realm.