The Vanguard rocket had three stages. The first two were liquid-fueled and the third was solid-fueled. That struck me as odd, as solid fuel engines are almost exclusively used in atmosphere, and you'd probably want the finest control for your last stage, when placing a satellite into orbit (especially one with no propulsion of its own).
What would have driven that design decision?
Solid fuel motors aren't unknown for upper stage use (e.g IUS, Star-48, among others).
While solids generally have the drawbacks of poor Isp and controllability, their simplicity keeps cost and stage dry mass down on small upper stages. Note that in early launchers such as Vanguard, the specific impulse of the solid stage wasn't much less than that of the liquid stages (248, 271, and 230 seconds respectively for the three stages); there likely weren't any very small, efficient liquid engines available for the third stage role.
Control of orbital insertion, also, wasn't that critical for the Vanguard series; the satellites were launched into fairly high, elliptical orbits, and it was challenging enough to get them up at all without worrying about their exact trajectories.
A solid stage can have a very low dry mass fraction, and can be spin stabilized avoiding the need for heavy guidance systems in the final stage. Both add mass directly to what is available for the payload. The Isp of solids is competitive with liquids other than LOX/LH2. A little bit of inaccuracy due to uncertainty in the total impulse of the motor can be made up by the spacecraft with a small amount of propellant. Overall a win.
The three-stage Delta II rockets used for many interplanetary and other missions in the 1990's and 2000's did the same thing, where the third stage was a spin-stabilized Star-48 motor.
New Horizons used a Star-48 motor as the final stage on the Atlas V-551 to get its high-speed boost to Jupiter and on to Pluto.
Because 1956 rocket scientists wouldn’t stake their careers on a liquid system not lighting. Same reason Atlas has a center engine, lit on the ground. Same reason R-7 Sapwood has a core stage, lit on the ground. A solid system lights just like a firecracker, candle, chemical generator, etc.: reliably.
Eventually, confidence in air-lit liquid stages reached the point of practice. But it took a lot of test runs, ground, then flight, then bigger flight.
