The school is TDY so you will return to your unit when you finish (hopefully as a graduate). From there, you will typically make an inter-unit transfer to the sniper section or platoon, depending on the type of unit you're in. Its not anything super formal, just a set of orders and an adjustment to a unit's roster.
You didn't mention what type of unit you're in (Cav, Airborne, Stryker, mech, etc.). Each of these units have organizational differences when it comes to sniper sections. Some maintain a larger section at the battalion HQ level only and will task out sniper teams to support company ops as needed, and some have small sniper sections at the company level (and battalion level as well) that are organic to that company. SF does things differently, they are typically cross trained in a variety of different skills and will support their team as they are needed.
I will echo the other sentiments on this post. Sniper school has small class sizes only a handful of times per year. Its a competitive school with a pretty high attrition rate. If you want it, you have to be a top performer in your unit, not just your company, but your battalion. They are only going to pay to send guys they know can make it so they aren't wasting slots, but even then, the school can still eliminate even the most qualified candidate. They also aren't going to send you to the school until you are competent in the basic tasks of your unit (whether it be infantry or cav). Volunteering for schools (even if they aren't shooting related) will also hep establish you as someone who can go to military schools and perform. Shoot expert on your assigned weapon consistently. High PT scores and performance during unit PT and on rucks/runs. Look at trying to get an intermediate shooting school such as SDM or LRM. See if the sniper section and your leadership will let you tag along to a range day. Trying to get into a specialized platoon (recon, RSTA, pathfinders for example). But in the end, you will likely be competing with several other people for slots, so you have to set yourself apart without putting your boot in your mouth. Many units may hold some sort of selection or tryouts open to the entire unit.
Above all, patience. For that type of school, you may have to wait years before you get a slot, if ever.