It was common practice in the advent of war to increase the military establishment. This could be achieved by raising new regiments, by adding companies to existing regiments, and by increasing the establishment of companies in existing regiments. This latter approach resembles best the idea to have a regular cadre force in peace time, which is used to build a new army in time of war.
Though England was at war with the Dutch Republic between 1672 and 1674 (Third Anglo - Dutch War, part of the larger Franco - Dutch War), the anti-Dutch feelings had changed into a pro-Dutch by 1677. There may be several reasons for this. One might be the marriage between William of Orange (future King William III (II)) and Mary Stuart (daughter of the future King James II (VII)). Also, the idea of a fellow protestant nation being conquered by a catholic nation (France) was not welcomed very much, which added to the general anti-Catholic sentiments in England.
Hence, England decided early 1678 to take the side of the Dutch Republic in their fight against France and the establishment of the army for greatly increased.
Apart for a dozen newly raised regiments, we also see the creation of second battalions to existing regiments. These second battalions should not be regarded as the implementation of second battalions in the later 18th Century, which had a full complement of regimental officers and staff. These 1678 implementations were first and foremost an increase of the regimental establishment by adding so many new companies, such that the regiment had the number of companies of two regiments, but without an additional lieutenant-colonel or major. However, a 2nd Adjutant seems to have been appointed. This, combined with references that the regiment was indeed split into two (battalions) for operations may be enough to conclude that these regiments were indeed multi-battalion regiments. Though in embryonic form of course, and the regiments should be viewed as large regiments with many companies that were subdivided, and not as a regiment with multiple battalions as we know it from later periods.
Only four regiments, at least on the English Establishment seem to have formed an additional battalion. The formation of the additional companies, and thus battalion, happened in January and February 1678. In early 1679 the augmentations were disbanded. These four regiments were:
Coldstream Regiment of Foot Guards
Formed 1650 in Parliamentary Army, and taken onto the establishment in 1661. Ranked after the 1st Regiment of Foot Guards.
The Lord High Admiral's Regiment
Formed late 1664 as a marine regiment, and was the highest ranking English regiment.
The Holland Regiment
Formed 1665, see this blog on the regiment's formation; ranked after the Admiral's Regiment.
The Duke of Monmouth's Regiment
Formed 1672 for French service as the Royal English Regiment, and returned early 1678 to England and placed 10 Feb 1678 on the English Establishment. In 1678 it is most likely that this regiment ranked immediately after the Holland Regiment. The first, original, battalion was styled Old Battalion, as it originated directly from the 1672 regiment. The second battalion was styled New Battalion. Though disbanded in early 1679, the regiment seems to have been re-formed in June 1679 following the Covenanter Rebellion.
The 1st Regiment of Foot Guards, the Grenadier Guards, already had a complement of 20+ companies. It is interesting to note that it were the highest ranking regiments on the English Establishment that were "elevated" to a larger complement of companies. The regiment of the Earl of Dumbarton, the Royal Regiment, seems to be on a complement equivalent to two battalions since the early 1670s, when in service of France and recruiting for the forthcoming war with the Dutch Republic. There was also a large regiment on the Irish Establishment, Thomas Dongan's, with a complement of 21 companies in 1678. This regiment was also formed for French service in 1671 and returned in 1678.