From what I can tell the progression of Chief Engineers by episode goes:
- Sarah MacDougal 1x03(The Naked Now)
- (Michael) Argyle 1x06(Where No One Has Gone Before), 1x13(Datalore)
- Logan 1x21(The Arsenal of Freedom)
- Leland T. Lynch 1x23(Skin of Evil)
- Geordi LaForge 2x01(the Child) onward
There are no instances where a Chief Engineer is mentioned, a new one is introduced, then an earlier one resumes the role. This progression makes me doubt the theories that there are multiple chief engineers to cover each duty shift but when Geordi takes over they reduce to one.
My theory goes that MacDougal was the Chief Engineer for the launch and initial shakedown, possibly she is a specialist in new ship shakedowns. Then was replaced by Argyle, who would last most of the first season, but then leaves, possibly retires, possibly reassigned due to personality conflict or inability to adapt to the new technology of the Enterprise (he never seemed overly innovative or anything more than competent). From then begins a cycling rota of "test" chief engineers from the ship's Junior officer pool. We see this in Arsenal of Freedom where Lieutenant Logan is less than secure in his position and threatened by Laforge's elevation over him, at this time Laforge is a command officer at navigation. Soon he is replaced by Lieutenant Commander Leland T. Lynch who is also proud of his recent promotion as seen by his insistence of responding with his full name when comm'ed. Just before s2ep01 "The Child" Geordi transfers to Engineering (first time we see him out of the red uniform) and he remains chief engineer.