The need for a turbofan was realized as early as the mid-30s.
The first jet aircraft were subsonic and the poor suitability of the propelling nozzle for these speeds due to high fuel consumption was understood, and bypass proposed, as early as 1936 (U.K. Patent 471,368). The underlying principle behind bypass is trading exhaust velocity for extra mass flow which still gives the required thrust but uses less fuel. Whittle called it "gearing down the flow".
In the mid-60s, the USAF A-X and CX-XX programs accelerated the development of the high-bypass jet engines. The winners of the programs were the Fairchild Republic A-10 Thunderbolt II and the Lockheed C-5 Galaxy, respectively.
To meet the [CX-XX] power and range specifications with only four engines required a new engine with dramatically improved fuel efficiency.
The TF39 on the C-5 Galaxy was the first production high-power, high-bypass jet engine. It first ran in 1964. The XV-5's lift-fan-inspired-demonstrator helped confirm to GE the feasibility of high-bypass jet engines.
The demonstrator was the GE1/6 (cruise fan variant), it featured a GE1 core engine energizing remote tip-turbine-driven cruise fan of high-bypass. (Sadly, I can't find photos/illustrations.)
Around the same time the Convair 990 Coronado was powered by the CJ805-23, which featured one of the first aft-mounted fans.
(Source) CJ805-23 with the aft-fan.
Transatlantic service needed a higher thrust version of the existing turbojet and a demonstrator engine with a single-stage front fan attached to the compressor was run. It was difficult to start and operate. This experience led to the aft-fan which didn't compromise the operation of the gas generator.
For its time, the TF39 had unique features—namely the 1½ fan-stage—which resulted in an impressive 8:1 bypass ratio, higher figures were seen only decades later on the Boeing 777's GE90.
The front of the engine features a mini-fan that rotates inside stationary vanes. Behind this first ½ stage is a full-sized fan. A fair proportion of the air entering the ½ stage is also ducted to the bypass stream.
(FLIGHT International, 13 March 1969)
The two-spool high bypass General Electric CF6 (top half section) is based on the same gas generator and turbine as the TF39 (bottom section) which powers the Lockheed C-5A Galaxy.
The mostly civilian and much simpler CF6 was based on the T39, it is found on the Airbus A300/310/330, Boeing 747/767, DC-10/MD-11, and others.
Two years before the A-X program, Avco Lycoming was working on the earliest experimental turbofans, but not necessarily high-bypass. With development and various derivatives contributing to the Lycoming ALF 502 which powered the Northrop YA-9 entry for the A-X program.
(FLIGHT International, 10 February 1972)
The only production fan engine to use an aft-mounted fan is the CF-700, power unit of the Fanjet/Mystere 20.
Sources: Various issues of FLIGHT, Wikipedia, and The Power to Fly: An Engineer's Life.