The quest for volume started in the 1890s when, influenced by the popularity of mandolin orchestras, guitarists began to replace their gut strings with wire.
The trapeze tailpiece was invented at the same time, to help the guitar's top support the increased tension created by steel strings.

Around the time of World War I, some makers began increasing the size of their guitars; witness Martin's first "dreadnoughts."

Dopyera Brothers - Dobro

A major breakthrough in the quest for volume was the invention of the resonator guitar by George Beauchamp and John Dopyera in 1927, now often generically known by the brand name “Dobro”.   

A resonator guitar produces sound by transmitting the vibration of the strings to an aluminum cone rather than to the wood top of the guitar. However, the Beauchamp and Dopyera resonator guitar was only the latest, and ultimately most successful, incarnation of experimentation that dates back to the mid-nineteenth century. Bob Brozman dates the earliest patent for an instrument based on the principles used by Beauchamp and Dopyera to England in the 1860s.  Beauchamp’s inspiration for the resonator guitar apparently was an amplified violin invented in 1899 by John M. A. Stroh. The Stroh violin was designed to produce enough volume to be captured by the recording technology of the era, which relied on sheer acoustic brute force to carve grooves in wax cylinders.  Stroh replaced the violin’s wood body with a metal resonator. The vibration of the strings was transmitted to a membrane made of thin aluminium, and the resultant sound then was directed through, and amplified by, an aluminium horn. The instruments were manufactured, beginning in 1904, by George Evans and Company, which also built a limited number of other instruments based on the same principle, including ukuleles, mandolins and guitars. The Stroh violin enjoyed modest success, but died out once electronic recording and amplification rendered it obsolete (except, for some bizarre and inexplicable reason, in a tiny region of Transylvania called Bihor, where it is known as the vioara cu goarnä and is still used in the region’s folk music). Stroh-style guitars never caught on, and apparently only three have survived. Other guitar-like instruments based on similar resonator principles were patented in the early 1900s by Alfred T. Bond of Rexburg, Ohio, and Samuel E. Buercklin of Prague, Oklahoma.

The Dopyera brothers' metal-bodied National resonator guitars presented the next solution to the volume dilemma.

Archtops Go Electric

While electric guitars first appeared following the invention of electronic recording in 1924 (Stromberg-Voisinet's Electro of 1928 is the first documented, if unsuccessful, electric), the next step in the evolution of the march toward volume was the predominance of the archtop guitar in the 1930s.

Paced by companies such as Gibson and Epiphone and individual luthiers such as New York's John D'Angelico and Boston's Elmer Stromberg, everyone touted their high-volume archtop guitars, including Stromberg-Voisinet, which had become the Kay Musical Instrument Company by the early '30s, and was heavily promoting guitars such as this swell Kay Violin-Style archtop guitar from 1938.

The guitar was a good candidate for amplification due to its acoustic properties and for its potential as a polyphonic solo instrument. The need for an amplified guitar became apparent during the Big Band era, as orchestras increased in size: particularly when guitars had to compete with large brass sections.

Alvino Rey

Joseph Reinhardt - with Grimshaw Archtop

The first Electric Guitars used in jazz were hollow archtop acoustic guitar bodies with electromagnetic transducers. By 1932 an electrically amplified guitar was commercially available.  A common mistake people make is thinking Gibson's ES-150 was the first electric guitar, but ES-150 was the name of the pickup, not the guitar. 

Alvino Rey was an artist who took this instrument to a wide audience in a large orchestral setting and later developed the pedal steel guitar for Gibson.

An early proponent of the Electric Spanish guitar was jazz guitarist  George Barnes who used the instrument in two songs recorded in Chicago on March 1, 1938, Sweetheart Land and It's a Low-Down Dirty Shame with Big Bill Broonzy. Some incorrectly attribute the first recording to Eddie Durham, but his recording with the Kansas City Five was 15 days later.

Durham introduced the instrument to a young Charlie Christian, who made the instrument famous in his brief life and would be a major influence on jazz guitarists for decades thereafter.

Eddie Durham  - He’d also been among the first to experiment with using an amplified guitar on jazz records. Leonard Feather wrote that Eddie Durham’s solo on Jimmie Lunceford’s “Hittin’ the Bottle,” recorded in September 1935, was “probably the first recorded example of any form of guitar amplification.” In the Eddie Durham article I co-wrote with Joel Siegel for the August 1979 Guitar Player, Durham explained that he began his experiments in 1929, while a member of Benny Moten’s Kansas City Band. “I got one of those tin pie pans and carved out my acoustic guitar’s top and put it down in there. It was the size of a breakfast plate. When you’d hit those strings, the pie pan would ring and shoot out the sound. I’d use a megaphone with it. I didn’t have that for too long, though, because I got a National steel guitar. It had a resonator in it, and it was usually played with a bar. I removed the bridge and put an acoustic-guitar type bridge on it so the action was lower. I fooled around with that for a long time, and nobody else was playing a guitar like that back then.”
Eddie Durham Medley

Big Bill Broonzy - It's a Low Down Dirty Shame
Sweetheart Land"/"It's A Low Down Dirty Shame", with Big Bill Broonzy, March 1, 1938, Acc. by Bill Owsley (ts); BJ Davis (p); George Barnes (el/g).

George Barnes (July 17, 1921 – September 5, 1977) was a world-renowned swing jazz guitarist, who claimed he played the first electric guitar 1931 aged 10years, preceding Charlie Christian by six years. George Barnes made the first recording of an electric guitar in 1938 (Aged almost 17) in sessions with Big Bill Broonzy .  At age 17, Barnes became NBC’s youngest conductor and arranger when he joined their staff orchestra. He was also given free rein over all creative endeavours. The network immediately put him to work on the Chicago based WLS and the NBD.