According to the "Dictionary of Christianity in America," the Puritans of the 1600's in America had a Sunday morning and a Sunday evening gathering time as well as a Thursday morning lecture. These practices almost certainly go back to the Puritans in England.
The mid-week meeting had its beginnings in prayer meetings that were occasionally mentioned before 1800 but became popular through the efforts of Charles Finney and D. L. Moody in the 1800's. Moody held noon prayer meetings in conjunction with his preaching campaigns. During the years 1857-1858, an awakening occurred that was later called the Prayer Meeting Revival. By 1900, prayer meetings or mid-week services became common in most evangelistic and many Protestant churches. By the mid-1900's, the prayer emphasis of the mid-week service was often replaced by a teaching or preaching service.
While the extra services are not commanded in scripture, they surely have its blessings. The apostolic church often met on a daily basis: "And daily in the temple, and in every house, they ceased not to teach and preach Jesus Christ" (Acts 5:42). Hebrews 10:25 tells us, "Not forsaking the assembling of ourselves together, as the manner of some is; but exhorting one another: and so much the more, as ye see the day approaching." Therefore, the closer we get to the end of this age, the more often we should meet together as a church. Perhaps it is time to add a new one. Thanks for this interesting question.