I would caution against the anachronistic use of the term 'trench warfare' in the American Civil War. The term is typically used to describe the Western Front of WWI, where the numbers, firepower, and dispositions of the combatants allowed the continuous front lines to extend from the Alps to the English channel. This did not happen in the American Civil War; the armies were not large enough to anchor their both their flanks on impassable obstacles, and did not have the firepower to prevent a frontal breakthrough with assurance. The technology of the two wars is very different, and they were fought by very different societies. The social-military conditions of the American Civil War were unique to their time and place, and should be understood on their own terms.

That said, there is indeed precedent for the use of entrenchments before the siege of Petersburg, really to the earliest days of the war. Really, it'd be easier to list the battles where no entrenchments were used. After the Battle of Bull Run, Joseph E. Johnston entrenched his army before he fell back to his base on the Rapidan Rappahannock river line; this position famously yielded the 'quaker guns' (logs painted to look like cannons) that baffled Union scouts and generals for months. Lee made extensive use of trenches during the Peninsula and Seven Days campaigns as an economy of force measure, allowing him to defend Richmond with minimal forces while he concentrated most of his army for the hammer blow to the north. McClellan was (in)famously trench-happy, establishing very strong positions to cover the slow, methodical advance to the next position, where he could dig in, bring up his heavy guns, and repeat the process. Joe Hooker had designed the Chancellorsville campaign around forcing Lee to attack his entrenched infantry, and Union positions on Cemetery Hill at Gettysburg were well fortified. The Overland campaign leading up to the siege of Petersburg saw field fortifications play a pivotal role in just about every battle. This is a very bumper sticker summary; I can go into more detail if you want.

However, like I mentioned above, I would not describe this as trench warfare, and I don't think the use of entrenchments during the American Civil War meaningfully anticipates the developments of the First World War. While the prevailing view is that the increased range of the rifle musket forced troops to seek shelter in the earth to survive, I agree with Earl J. Hess, who has authored a series of books on the use of field fortifications during the Civil War. He strongly denies the claim of increased firepower; the available evidence indicates that engagements largely stayed within the range of smoothbore muskets, and that casualty rates were broadly similar. The necessity of accurate range estimation to hit anything past a hundred yards and the near total lack of marksmanship training for the rank and file severely limited the application of the new weapon.

Rather, the America proclivity towards entrenchments arose from a variety of less material causes. In the east, the inability of either army to destroy the other in a battle of annihilation meant they remained in contact much longer than European armies of the same period. Moreover, the prewar education of the engineers, who comprised the best and brightest of the pre war Regular Army, was directed by Dennis Hart Mahan of West Point, who placed great emphasis on the power of field fortifications. Mahan understood that American armies would largely be composed of militia and short service volunteers, whose forbearance under fire and the press of cold steel would be uncertain, and that prepared positions would be useful for stiffening otherwise shaky troops. The inverse of this turned out to be that said shaky troops would become very reluctant to assault field fortifications; by the time of the Overland campaign, 'worn out' regiments could flat out refuse to assault entrenchments, or make only a token effort.

For the full story, read Earl J. Hess's three volume history of entrenchments in the Civil War.

Field Armies and Fortifications: The Eastern Campaigns, 1861-1864

Trench Warfare under Grant and Lee: Field Fortifications in the Overland Campaign

In the Trenches at Petersburg: Field Fortifications and Confederate Defeat