It is not true that condos do not have board packages and that this can be done in days. In some condos, yes, in others, no. I just represented the buyer of a condo who needs to rent it out for the first year. Right after we closed, he signed the rental listing agreement and I had a very qualified renter in a couple of days. We now have to do the condo board package which is essentially identical to the purchase package we just did. The time frame depends upon the building. So, to answer your question of how to keep the gap as short as possible between closing and renting, I would say check with the managing agent now on what the sublet application to the board is and get a copy right away. Then line up who you will list the rental with. I am assuming you used an agent, and you will probably want to stick with that agent if they served you well. If you did not use and agent, line one up now for the rental. The day you close, sign the rental listing agreement. If you have all that lined up, the gap will be based upon how extensive the sublet board package is and how long the managing agent and the board take to review it. It varies greatly.
Halstead Property, LLC
http://jenetlevy.halstead.com for all NYC listings
If I were you, I would get the help of a real estate agent. You want someone who will provide a legal lease, all the required disclosures, obtain credit checks, etc. As a former journalist I personally verify employment and references of tenants for my clients by phone, I don't rely on paper information. Feel free to contact me if I can help.
Licensed Real Estate Salesperson
New York, NY
The renting part is easy with a great agent.