The only U.S. built rocket or missile that I can think of that dropped a tall section off was the Atlas. Launching with three engines, then when it was light enough to only need the thrust of the center engine, dropping the outer two inside of one assembly. Referred to Stage-and-a-half. Part of the reason for that were the problems of igniting liquid upper stages.
Certainly the Redstone did not drop its tail unit.
What the wiki actually says:
. The missile body was separated from the thrust unit 20–30 seconds after the termination of powered flight, as determined by the preset range to target. The body continued on a controlled ballistic trajectory to the target impact point. The thrust unit continued on its own uncontrolled ballistic trajectory, impacting short of the designated target.
The wiki is referring to what we think of as the WARHEAD SECTION separating, from the Redstone Rocket booster. "Missile body" is somewhat misleading to us now but in the drawing below the part containing the "Warhead section" is listed as "Body" (on the drawing at right) versus the rocket portion which is what we would consider to be the Redstone ROCKET, is referred to as the "Thrust Unit" on the leftmost drawing.
The separation point was between the Thrust Unit (entire rocket booster) and "Aft Unit" which had the air vanes to help steer the warhead section during the downward leg of the trajectory (of course missile accuracy in those days was very poor, this short range missile could easily miss by a mile or more. And in many ways this was more like an updated and uprated V-2).
This is the problem of reading details about one version of a rocket when there are significantly different versions (Jupiter-C and Mercury Redstone) but the author does not clarify what applies and what does not. Of course that wiki page was about the missile version. And while the author seems to have been historically correct in referring to the Redstone rocket portion the "Thrust Unit", it is too misleading, as seen in posts above. I mean, "THRUST unit" being the whole rocket body, while "TAIL unit" is the portion that actually had the THRUST producing device (engine) inside of it.
So I can't fault the author too much for use of "historical" terms, but the author should have realized the potential for confusion. But when he referred to the separated front section (Body, warhead, whatever) as the "missile body", that really messed things up. I cannot think of any military missile where the whole rocket was not called a missile, and only a separated front end with no rocket engines was referred to the missile instead of the whole rocket that launched it.
So, on the MERCURY Redstone, the only things that separated were the Escape tower from the capsule, and the Capsule from the Redstone rocket. Well, and the retro-pack after retrofire.
FWIW - drawing of the three Redstone based rockets. Jupiter-C (aka Juno-I later) also separated from the top end of the Redstone body and had steering thrusters to aim the upper section to the correct position to fire the spinning 2nd/3rd/4th stages to put Explorer satellites into obit. While the Mercury Redstone, the "instrument unit" stayed attached to the Redstone.