After promotion to SSG and completion of SSD3, when would you recommend a soldier submit a 4187 to attend SLC?

Read MILPER Message 16-089. That explains STEP NCOPDS Scheduling. I'll help you out by posting the SLC portion:

5. SENIOR LEADER COURSE (SLC) SCHEDULING:

A. SOLDIERS WHO ARE SELECTED BY THE CENTRALIZED PROMOTION BOARD FOR SFC WILL BE SCHEDULED TO ATTEND SLC BASED ON ASSIGNED SEQUENCE NUMBERS BY MOS. SOLDIERS WILL BE SCHEDULED REGARDLESS IF THE UNIT IS ON THE PATCH CHART. IF COMMANDERS DETERMINE A SOLDIER CANNOT ATTEND TRAINING, A DEFERMENT REQUEST MUST BE SUBMITTED TO HRC IN ACCORDANCE WITH AR 350-1 AND THIS MESSAGE.

These are the priorities for how SSGs are scheduled for SLC:

7. SCHEDULING PRIORITIES:

A. PRIORITY 1: BEGINNING 1 JAN 16, PROMOTABLE SOLDIERS WITH THE HIGHEST PROMOTION POINTS TO SSG WILL BE SCHEDULED FIRST FOR ALC AND SOLDIERS SELECTED FOR SERGEANT FIRST CLASS (CY16 LIST) WITH THE HIGHEST SEQUENCE NUMBER WILL BE SCHEDULED FIRST FOR SLC.

B. PRIORITY 2: ALL PROMOTABLE SOLDIERS IDENTIFIED FOR DEPLOYMENT, SHORT TOUR, OR UNIT ROTATION PRIOR TO THE EVENT; PROVIDED THE SOLDIER WOULD HAVE OTHERWISE BEEN SCHEDULED DURING SUCH DEPLOYMENT, SHORT TOUR OR UNIT ROTATION. IF UNABLE TO DO SO, THEY WILL BE SCHEDULED FOR TRAINING AT THE END OF THEIR STABILIZATION PERIOD OR UPON REASSIGNMENT BASED ON THEIR PROMOTION POINTS OR SEQUENCE NUMBER AND MOS.

C. PRIORITY 3: SOLDIERS IN THE LEGACY BACKLOG DEEMED AT RISK FOR QUALITATIVE MANAGEMENT PROGRAM (QMP) (37 OR MORE MONTHS TIME IN GRADE).

D. PRIORITY 4: SOLDIERS IN THE LEGACY BACKLOG WITH LESS THAN 37 MONTHS TIME IN GRADE AND SOLDIERS IN REGENERATION UNITS.

E. PRIORITY 5: TRAIN AHEAD SOLDIERS (NOT IN A PROMOTABLE STATUS) THAT CAN BE SCHEDULED FOR CLASS THAT DO NOT MEET PRIORITY 1-4 CRITERIA TO ENSURE CLASSES ARE CONDUCTED AT OPTIMIZED CAPACITY. REQUESTS TO TRAIN AHEAD WILL BE SUBMITTED THROUGH THE CHAIN OF COMMAND TO HRC. ONCE THE CHAIN OF COMMAND PROCESSES THE REQUEST, HRC WILL SCHEDULE THE SOLDIER BASED ON AVAILABILITY OF TRAINING SEATS. HRC WILL SCHEDULE SOLDIERS IN MOS’ WITHOUT PROMOTABLES BY DATE OF RANK (DOR).

That's how people get scheduled for SLC.

I'll tell you this. I had to defer SLC the last two years because of my daughter's severe medical issues. I went to try to reschedule for January 2018 because, although she's still serious, she's stable enough on dialysis to stay with my family. I did what HRC said in getting a memo signed by a GO. They came back and said ""The
Soldier will be scheduled IAW STEP."

That's how people get scheduled these days it seems. So if someone has just made SSG they aren't going to SLC any time soon. Priority goes to a SSG (P) with a sequence number.

Its great your thinking about this but you can not attendt SLC until yoy have been selected for SFC promotion this all part of the Armys S.T.E.P program this has been around for a while now!

If you are trying to get promoted with your peers wait until they tell you to go. If you want to pass your peers. Prepare the 4187 and submit it. Go immediately. Its a school.
You only have to ask yourself what you want out of the Army, no one elses career path matters when it comes to yours.
Take care of your Soldiers and take care of yourself. If you do that the mission will be successful and so will you!