[support] repairing a website

I have inherited a website that has third-party modules installed into the modules folder at the root level with the core modules, not sites/all/modules folder. Initially, I tried to move the directories one at at a time into the appropriate folder and the site broke. The core is 6.19. Would it be best to upgrade the core first? Should I disable the module, followed by downloading and installing the same module in the correct folder and then re-enabling it? Should I download and install all of the third party modules in the correct folder first, delete the other inappropriately located folders, and then run update first? Do I need to recreate the entire site from scratch?