First and foremost, I admire you for taking care of your daughter and starting over in Florida. I had a similar situation where I had to take a year off work just to prove to the courts I was as capable as her mother to care for a child. Go to http://www.freecreditreport.com
and add a Statement to your report stating the reason as being a divorce and that you take care of your daughter. You won't have to put up a "higher" security deposit but you will have to put up first, last and security. Be up front and explain to the potential landlord your situation and stress your integrity, professionalism as a chef and responsibility for taking care of your child. Renting a property is like a job interview. You need to assure the landlord that you are a stable, honest person that will pay their rent on time and not destroy their property or create problems for them. You may also want to think about moving to Naples or Estero Florida. Low rentals are available in these areas also now and Naples is a lot safer than Ft. Myers as far as crime goes. Also, Estero is a booming area due to the newest University, Florida Gulf Coast University. Naples is the most pristine city on the Gulf Coast in my opinion. I've lived all over the world and ended up coming back to Naples to raise my daughter. In your line of business there are many outstanding restaurants to hang your hat. Some of the best restaurants are Handsome Harry's, Chops City Grill (a Naples standard), and Trulux. There are many, many more. People eat out a ton in Naples and they still have the money to do so.
You may also want to contact a rental agency, I assist some clients, that caters to relocation and annual rentals. Explain your situation but stress your ability and willingness to pay on time and your professionalism. You should be able to find a great rental and home to start your new life for you and your daughter. Visit my web site to see rentals (choose "Rental" in the Property Type drop down menu) in Naples and Estero Florida. Best of luck to you and your daughter! - Mark Weber (239) 289-0799