Comparing risk of event in a subsample versus the entire sample (with overlap)

I have a relative risk (RR_homeless_SMI) that provides me with the likelihood of becoming homeless in people with a serious mental illness (SMI) compared to those not having an SMI at all (NO_SMI).
I also have a dataset where I have frequencies on homelessness in people with an SMI (in remission [SMI_REM] or relapse [SMI_REL).
Remission or relapse status are defined by a clinical score.
What I ultimately need is to be able to calculate the likelihood of being homeless in people with a SMI_REM or SMI_REL departing from the probability of homelessness in people with NO_SMI.

Is the below appropriate?
pHomeless_SMI_REL = pHomeless_NO_SMI * RR_homeless_SMI * RR_SMI_REL
RR_SMI_REL = pHomelessness_REL/pHomelessness_SMI

Can I calculate a relative risk (probably this is not the right denomination to this ratio) of part of the sample divided by the entire sample (with overlap)?
How do I calculate the confidence interval for this ratio (RR_SMI_REL)?

Any insight would be much appreciated.