It sounds like you want the usual situation to be the self consistent time loop, but under certain extraordinary circumstances you want the characters to be able to change events, perhaps (somehow) to different self consistent loop.
This is very similar to the concept of a strange attractor.
In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system.[1] System values that get close enough to the attractor values remain close even if slightly disturbed.
I'm imagining a metaphor for the time loop as a ball rolling around in one of various circular valleys. Some are of the valleys are deeper than others. Some valleys are bigger than others. 419
The interesting case would be the valleys that nearly touch another, and so at a key moment it might be possible to make just the right change to switch from one valley to another, and so from one time loop to another.
A planet that orbits around one star in a binary system, and whose orbit is close to the null point between the stars does something like this.
I would use the "sciency" phase "quantum bias" to describe a tendency for the effects of most small changes to not create butterfly effect ripples, but for the odds to usually stack themselves so that even with big choices and big changes, that events will tend to counterbalance and turn out in almost the same way.
The question would then become, "What sort of key point in the time loop is most likely to cause the loop to shift from one to another?". Most time travel stories will have something irrevocable (like a death preventing a birth) be the event that tips things to a different sequence. I think it would be fun if the event was deeply random, causing a <PERSON>-Goldberg-eque cascade of events.
The most obvious solutions to how just the right event could be found (they should be rare) would be either a quantum artificial intelligence, or someone with a mysterious talent/sensitivity. Or more comically, you could have a "anti-Gilligan" who through dumb luck tends to stumble into the event that will shift the situation to the new loop. 239