re But I'm not sure what I would add
The first thing I would do is make sure that the claims (at least in form of dependent claims) provide support and enabling for this "encapsulated idea". That way, if the examiner feels that you do not have support in the original specification, you do have such support in the claims themselves. You can then rename the "continuation" a "continuation in part".
I have had this happen to me - I filed a continuation with new claims, the Examiner said I did not deserve the priority, but he issued the case anyway.
There is the risk that the new idea be considered obvious (but not described). The idea behind a CIP is that the independent claim is clearly supported by the earlier specification; this reduces the risk, but does not make it go away. Clearly supported = uses literal language from the original filing.
Anyway, if the idea is "encapsulated" this suggests to me that the idea and its useful implementations are not well described in the original application, so that is what I would add in a CIP. During examination, I suspect some claims will get the earlier date, some will get a new date and some might be indeed obvious.
Having said all the above, I am generally against CIPs, as, usually they are filed more than one year after publication of the earlier application, which allows the earlier application to be used for obviousness. So if you have a new idea, start with a new filing and if the idea is described, use a plain continuation.