How long does one have to be in a support role before gaining the “S” SQI?

It doesn't mean anything and doesn't do anything for you. USASOC positions are filled on a first come first serve basis for lower enlisted and a slating board for senior enlisted.

There is no such thing as a lateral move to a USASOC unit on AD, even when it's the same post it is a PCS because you're moving outside the MACOM. That means HRC has to place you on assignment to go there. The often referred to "Inter post transfer" to Group isn't actually how it works.

I'm not entirely sure if your time with 20th SFG really counts toward crediting you with an S SQI but in order to receive your S qualifier you would need to currently be assigned to a USASOC position, be airborne qualified, and be recommended for it by the commander.

Again, it does nothing for you or your career. It won't get you to a group if there's no slot, and you don't need it to get slotted there. It won't get you selected for an assignment over another person.

USASOC, in coordination with HRC, developed the skill identifier K9 (special operations support) and the special qualifications identifier S (special operations support personnel). These identifiers are for individuals who have successfully served 22 months in a SOF formation or 12 months in a SOF deployment.


https://www.army.mil/article/166176/supporting_army_and_joint_special_operations_forces

Thanks for the question Connor