You get one puny picket at size 3 patrol HQ (assuming focus on quality). If you want bigger fleet, you need to grow your colony (use Free Port and/or Population Incentives). Fleet Logistics 3 will help fleet size too. By size 4, I might have had two smaller pickets. By the time I got to size 5, I had boosted ship quality (with Orbital Works and corrupted nanoforge), allowing me to shift doctrine from quality to more ships a bit, and I got a military base and a battlestation. I got more and bigger fleets. By size 6, my patrols are about on par with a major faction's. I routinely see two or three patrols - my patrols - flying around my system.