I was in your shoes a few years ago. Enlisted SB and DOR'd. I re rated to AE and made the best of it. I knew I wanted to drop an OCS package as soon as I got to A School but timing and Chain of Command really play into this one. I don't believe there is a strict time limit on as to how soon you can submit one but you need to consider that you are now obligated to serve in your rate for whatever length of time you signed on for. You still need to perform at A school C school and your first command. If you cant do your specific job well this will not look good on you and you chain of command will have a hard time getting behind you to support your request.
You're going to need time off work and/or a flexible LPO/Chief to let you schedule your interviews with current Officers from the field you're looking to get in as well as time off to get medical appt set up for the commissioning physical and they probably wont take you seriously until you get your quals and have proven your worth in the shop.
Bottom line here is that you need to prove yourself in your rate first before you really, really have a good shot at a quality package getting sent in.
I advise you to not lose sight of your goal, strive to make rank quickly and take on responsibility early. Even if it is outside the normal scope of your duties. These qualities well reflect well on you and when you go to ask your COC for time off or help with your package they will see you are the real deal and a proven leader even on a micro level. Warfare pins will come, be hungry for quals. Quality over quantity. Don't just get every qual in the shop because you want to be a paper tiger. Get what is required, get good at it then branch out.
I enlisted 2012 under an SB contract. Rerated 2012, E3 Aviation Electricians mate 2012 and made AE2 by May 2014. After 3 OCS packages got picked up and started OCS and commissioned 2015.
Its not impossible. Stay determined and feel free to message me if you have an questions with your process. You'll find out that applying from the fleet, not a whole lot of people have the answers your looking for when it comes to "who do I talk to next?" or "how is my application supposed to look?".
-Logan