Are there no episodes that dealt with the events leading from Earth finding how to warp?
No. Nothing in Star Trek at the highest continuity level (e.g. official, filmed Trek) discusses the FC to Broken Bow era.
If you include TAS along with Voyager, Enterprise and TNG (some from a visual display in Up a Long Ladder) you get ...
Zephram Cochrane's first warp flight as on April 5, 2063
The SS Valiant left Earth relatively soon after, eventual they reached the galactic barrier, in the year 2265 the ship had been missing over
In 2067 the warp drive probe Friendship One leaves Earth, eventual it reached the delta quadrant.
Around 2069 the Human colony bound for Terra Nova left Earth, arriving on June 23, 2078 after a nine-year warp flight. By the year 2151 the colony had been silent for over seventy years.
The Terratin colony also left Earth around this time, they originally named their colony Terra Ten. Suggest a number of colony efforts before their departure.
By the year 2102 Humans were building multiple J-class freighters (where Travis was born). Also in that year the SS Hokule was on a mission of "deep space" exploration.
In 2105, the VK Yuri Gagarin was launched from Earth on a colonization mission.
In 2120 the HMS Lord Nelson (British?) was launched from Earth on a mission of "deep space" exploration.
In 2135, the HMS New Zealand was launched from Earth on a diplomatic mission.
In 2146, the Urusei Yatsura was launched from Earth on a mission to conduct a nebula survey project. In real life the closest nebula to Earth is SH 2-216 which is about 400 light years away, Helix Nebula at 700 light years is the closest of the bright planetary nebulae.
In 2151 Travis said: "I always thought lost colonies affected boomers." Travis statement was with "colonies" in the plural, so Terra Nova was not the only lost colony. This implies Earth had multiple colonies by 2151.