from the SemVer spec https://semver.org/
A pre-release version MAY be denoted by appending a hyphen and a
series of dot separated identifiers immediately following the patch
...Pre-release versions have a lower
precedence than the associated normal version
... Precedence for two pre-release versions with the same major, minor,
and patch version MUST be determined by comparing each dot separated
identifier from left to right until a difference is found
So it seems you can have 1.1.1-rc1 and 1.1.1-rc2
To be honest though, I think this pre-release stuff makes a mockery of versioning and testing. Release or don't release 1.1.1 and follow with 1.1.2 if you fix a bug.
If you have more than one supported version then make sure they are different major versions or you will need the fourth version number.
I think the question has been edited a bit since my answer.
re: what should I call v1.1.0 + urgent bug fix, when 1.1.1-rc1 is already in the pipeline but wont be released quickly enough for me?
Well, there's no rule that you have to release every version, 1.1.1 has failed testing, because you know know about this urgent bug. release 'v1.1.0 + urgent bug fix' as 1.1.2-rc1 and then 1.1.2 as per your normal procedure.
I can see your objection would be 'but we are talking about 1.1.1 having a new feature, releasing 1.1.2 without the feature will confuse customers and reveal that we missed this bug!'
To which I say, if its a new feature it should at least be 1.2.0 if not 2.0.0! the third digit is for patches, not features. Do the feature as 1.2.0, 1.2.1... etc and you can continue with 'urgent' fixes without the feature on 1.1.3, 1.1.4 etc.
I know some companies are contractually obliged not to update the major number, if you are in a similar situation perhaps you need to add the forth version number and have 126.96.36.199?
Don't get into the situation where there is fear about upping the version number to correctly indicate different builds. Someone somewhere has that version 1.1.1 you released by mistake and needs 1.1.2 not 1.1.1 (corrected, hope no one notices!!)