There is a posted process listed on faajobs.gov.
If you want to actually be hired, you MUST know someone at the FSDO you want to work at. Complete the online applications, print out extra copies of the results, have someone walk in your resume to the manager responsible for the vacant position, get called for an interview, present your printout from the website to the manager who can then request you be hired through regional.
Many ASI jobs are filled before the listing ever appears online--got to know someone inside the FSDO to even know they are, or might be looking.
No one gets hired after July. No money in the budget! October 1st is often the start date, which may be delayed based upon FAA reauthorization. It was last time.
All budgets are controlled by region, so just because they want you and have an opening doesn't mean there is money to pay you!
If you apply online only, some clerk in region decides if you get to be part of a list of 10-20 names forwarded to the district office. Often times, that is just for show, they have already picked whom they want via the backdoor process I described.
Every office, district, and region is somewhat different. Your mileage may vary.
--Inconceivable, Air Carrier Assistant Program Manager, xxx FSDO, 2002-2006.