Well, this is not a very simple question. My personal view based on the research of the speedster historians that I know and the old pictures I have seen, I think the phase had pretty much run its course by the Model A era. When you ask folks for Model A speedster pictures, no one seems to be able to find/produce any pictures of an actual speedster built during the day.
In the truest definition, a Model T/A 'speedster' really did not use a body that Ford built. By that, if you go back and look at old pictures, many T speedsters used seats, fuel tanks, etc. from other marques of cars to make a vehicle that did not look like a Model T. Also, in very rare instances were the chassis' underslung or had axles dropped, or had the frames Z-ed simply because the person(s) building the speedster generally just did not have the budget nor the skills.
Therefore when you think about the Model A era, contrary to what some would have you believe, people did not go down and buy a new Model A and convert it into a speedster. Therefore someone needed to find a used car, which during the depression era still brought good money in a time where there was little extra money. Therefore speedsters just weren't built. After the depression, everyone wanted a new car, ...which by now were V8 Fords and 6 cylinders (other marques) that had more power. Therefore if a kid wanted a fast car, he would buy a 5 year old flathead Ford instead of a Model A. If he was broke, he bought the Model A but did he have money for power adders (speed equip. as we call it today)? Also when you speak of speedsters, generally these were vehicles without tops or a means to protect someone from the weather. By the late 30's, the trend had pretty much done away with open vehicles like roadsters & phaetons simply because people didn't like being cold or wet when they drove somewhere.
Another use for an old Roadster during the late 30's and 40's was to build a race car and run it at the county fairs in what was called Roadster Racing. I have included a few pics below depicting what these were however if you notice, by the truest definition, these aren't "speedsters" because they are still using the factory body. It is also worth noting that speedsters were generally patterned after what was the trend of car being raced at the time. At Indy and other tracks, the boattailed look had dissappeared by the early 1930s, and in 1940 it would have been very unlikely that people took an old Model A and bought a 1939 or 40 Ford hood to make them a "boattailed speedster". Could it have happened? I suppose there might have been a slim chance, ...and I guess that is why some people like to "re-write history" where it will apply to their desires. THAT is why I don't think there were hardly any Model A 'speedsters' but there were lots of Model A hot rods, gow jobs, etc.
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