You can pretty easily get to 2nd LT in a day, it will go a little faster if you have some stuff to turn in for seals (you can do at Sgt 2nd class), you figure this will open your new GC hunt log which also gives nice amount of seals too.

Since you are already 1st LT, your squad keeps their stats, so once you get them opened again, you only need to send them out on the promotion mission again and you are done ! ( I just went from Serpents to Maelstrom myself )