As others have mentioned here, there used to be something called "FX" motors which were basically model rocket motor-sized casings filled with "smoke bomb" propellant... you ignited them with a separate launch controller firing circuit just like you would a model rocket motor, then after a couple seconds ignited the regular model rocket motor to propel the rocket, assuming a small plane hadn't appeared directly over your launch pad or the pad blew over and the rocket was pointed right at you. They were never terribly popular and didn't last but a few years before production ceased and they disappeared. They were nice because they also augmented the smoke trail IN FLIGHT...
The motor manufacturers just plain WILL NOT modify their motors to include a second or two of delay-train propellant (which is slow-burning smoke producing propellant, compared to the fast-burning main propulsion propellant ignited at liftoff) for two reasons. 1) it would require a complete recertification of the motor, which is expensive and time consuming, and of course in addition to the redesign and testing of the motor in the first place, for a VERY limited market potential. 2) Many motor designs simply DO NOT have the extra room in the casing for the additional layer of "delay train" propellant to go in the nozzle end of the casing first, with the fast burning propulsion grain above it, topped by another delay grain on top of that, capped by the ejection charge and clay cap. Some motor designs aren't even feasible anymore because modern BP is sorrier than BP used to be, meaning that the propellant grains have to be longer than they used to be to get the same energy, which makes the entire slug too long to fit in the existing casing. Never mind adding ANOTHER layer of smoky delay propellant for special effects.
Now, there's been a number of ways to create the effect your talking about. Some of them fall outside the safety code and cannot be discussed in an open forum... PM me if you want more information. Personally I used to have a launch pad with a "blast tunnel", constructed of 3/4 inch plywood layered to give a base, with the tunnel lined with cut sheet metal, flanked by a "C" shaped piece of plywood to create the tunnel walls, and topped by another sheet of plywood with a hole cut in it over the blast tunnel to create the "flame hole" the rocket motor sat directly over. When I wanted lots of smoke, I simply dumped a few tablespoons of flour into the tunnel before I launched, which the rocket motor's blast would blow out of the tunnel at liftoff, creating a cloud of "smoke" as the rocket lifted off. Dr. Zooch built a "milkstool" launcher for one of his Saturn IB's similar to that used to launch Saturn IB's off Saturn V pads for Skylab missions and ASTP... he made a "smoke effect" by simply "CHAD staging" a dash zero booster motor directly to the motor in the Saturn IB, with the booster motor securely attached to the launch pad to prevent any movement, which was then ignited electrically like any other rocket motor. The booster motor burned "on the pad" and blew out copious clouds of smoke, and then at burn-through ignited the regular model rocket motor in the Saturn IB, "staging" on the launch pad, releasing the Saturn IB to then take off normally... I've heard of some other ideas, but they're definitely in the "gray areas" of the safety code at best, so if you want to know more you'll have to PM me...
Later and good luck! OL JR