In your shop manual you will find descriptions of 4 different manual transmissions used for a '66 Mustang. The 6s used the 2.77 3-speed while the V8s used the 3.03 3-speed. I think those numbers are the distance, in inches, between the main and lay shaft inside the transmission. For the V8s the 4-speed was officially called the 'Ford' 4-speed but is now usually called a Top Loader since the plate that opens up the gearbox is on top. V8s could also have had a T-10 Borg-Warner designed 4-speed since Ford couldn't make their own fast enough. The T-10, usually with different gears and minor other differences, was used in many other cars of the day including the Corvette. You won't find the T-10 in the '66 Mustang shop manual but its in the '65 Mustang shop manual (the '66 design was somewhat different but the exterior was the same). The 4-speed used for a 6 was the Dagenham which is the name of the eastern London, England, suburb where those transmissions were made by Ford of England.
As mentioned, every one of them had the exact same gear ratio in high gear: 1.00. Once in high gear your engine will act exactly the same no matter which of those transmissions you use. If there is a problem with how your engine behaves in 3rd gear its an engine problem and nothing to do with your 3-speed transmission.