What we did for this case was to "not break the API during a minor version".
For instance, only return more data and not less. Do not change the conditions/meaning of errors.
If you need to, then it's the right moment to switch to v2.
Exactly. All minor/revision API changes should be backwards compatible. You can add new optional parameters and new methods/objects, but you can't change the semantics of anything existing.
If you need a breaking change, that's time to increment the major number. Besides that, who'd want to support that many versions running live?
If you need to, then it's the right moment to switch to v2.