The question can be divided into two parts actually:
- How to trigger an Industrial Revolution, and:
- How to find cheap energy in post apocalyptic environment?
Question 2 can be answered in a multitude of ways. After 600 years, much of the forest cover of Earth would have regrown, and wood can be accessed directly as firewood, converted to charcoal for more energetic fuel (charcoal is the fuel of choice for industrial processes like smelting pre coal, and in the modern era, we convert coal to coke, which serves a similar purpose). For higher energy portable fuel, wood can be converted to wood gas, and cars were powered this way during WWII. So woodlots and careful forest management can provide the fuels needed.
On the question of founding a new Industrial Revolution, there are still lots of questions why the first Industrial Revolution happened.
The Ancient Greeks and Romans had plenty of machinery. Hero of Alexandria wrote a book on mechanisms we would identify as low powered steam and atmospheric engines, and the Romans knew of machinery like cranks to convert rotary motion from waterwheels and the like to linear motion, and items we would recognize as clockwork. The Antikythera mechanism is an elegant hand made analogue astronomical computer, and there is no reason to think it was the only one ever built (although there is no reason to suspect they were common either). Despite being decent engineers and having the ability to create and understand machinery (torsion powered Ballista are actually elegant and sophisticated machines), neither the Greeks not the Romans ever embarked on an Industrial Revolution.
Modern reproduction of the Antikythera mechanism
Torsion powered Ballista
In the Middle Ages and early Renaissance, nations as diverse as the Hanse and the Serenìsima Repùblica Vèneta developed sophisticated social and economic institutions. Venice even has a prototype assembly line (Arsenale di Venezia). However, despite these advances, neither the Hanse nor the Republic made the move to an Industrial Revolution. The origins of the Industrial Revolution in England are most likely explained in social or organizational terms, since many of the factors in England existed in other nations and other times. Even the handwave of suggesting the Steam engine and coal mining isn't quite enough, people have mined for millennia, and both the Ancient Romans had a form of steam engine, and the modern steam engine was patented in Spain by Jerónimo de Ayanz y Beaumont in 1606 (also for the use in draining mines).
You could in fact end up with a society using hand built wood, charcoal and coal gas powered machinery. This society could even be relatively sophisticated, with long distance trade and a relatively high standard of living, yet not having triggered an Industrial Revolution.