In my experience, following a major schedule change, most airlines will happily permit you to readjust your itinerary as you see fit, for free, regardless of the original fare rules.
For instance, that may include travelling earlier or later, the insertion or removal of a stop or layover, rerouting via different points, a change of origin or a change of destination, a change of flight on the same day, et c.
That freedom tends to be circumscribed as follows—
- You can usually only reroute on the same airline, or if airlines are in a close grouping, on those airlines. For instance swapping between British Airways and American Airlines going over the Atlantic is no problem, but swapping from Air France to rival carrier Lufthansa is not going to happen.
- If the city is no longer served by the airline, specific guidance for customers in your situation will be provided and may be quite restrictive. Usually that means interline carriage on a local carrier or a refund only.
- You can only change your itinerary once per schedule change. So try to get it right.
- The airline will offer you what they think is the best option. You need to be proactive and say "no, this doesn't suit me, I would like to book this alternative instead".
- The schedule change has to be significant (as a rule of thumb, over two hours, but it varies by airline and sometimes it is up to the agent's discretion). A cancellation forcing a reroute is always significant.
- You can't be totally unreasonable, it has to be broadly similar to your original ticket.
- Each airline has its own rules for exactly what changes you can make, it pays to be aware of this in advance.
My experience has been built up from exactly the situation you have. I buy a lot of fares that prohibit stopovers altogether, or would incur large additional taxes by including them (such as UK air passenger duty, currently an eye-watering £150 for a stopover). I also like to nest lots of tickets together, and if one ticket in the middle is changed it can ruin the timing of the other tickets. So I suffer at the hands of schedule changes and cancellations. They happen. But I have never had an agent say "our contract is A to B, we wash our hands of your stop at C"—rather, it is more along the lines of, "yes sir, your proposed rerouting is no problem, I will send it to re-ticketing, sorry again about the inconvenience".