If the organization is still a Connecticut corporation (simply admitted to do business in Florida) then the Articles of Amendment to its Articles of Incorporation need to be filed first in Connecticut, then in Florida.
On the other hand, if the corporation was re-domesticated in Florida, then the amendment should be file in Florida.
I suggest you check the corporation's name on the Connecticut Secretary of State's website, to see if it is still incorporated in Connecticut. Depending on what you find, proceed accordingly.
It might be both simpler and less costly to have a corporate attorney handle this for you.
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The process of changing your company's name is going to have two stages. First, you will need to file a certificate of amendment to the existing articles with the Connecticut Secretary of State. I would recommend, if you decide against using an attorney, at least to work with a filing agent. They are not lawyers, but they know the ins and outs of state filings and can be a real asset.
Once you finalize the amendment in Connecticut, you have to amend your registration (qualification to do business) in Florida. There is usually a simple form to fill out and the Secretary of State website should have forms and information. Again, working with a filing agent or an attorney will save you a lot of time and energy.
Please note that prior to changing the name, you should check both states for availability of your desired new name!