Extensions of CIR

I could need some advice on extensions of the CIR model.

$dr(t)=\kappa(\theta-r(t))dt + \sigma \sqrt{r(t)} dW(t)$.

A possible extension, if we would like the short-rate to also include negative values, could be a displaced version, so that $r(t)+\alpha$, where $\alpha>0$, follows a CIR model.

Further, to fit the initial term structure one could also consider the CIR++ (can be seen in Brigo et al) which is that

$r(t)=x(t)+\phi(t)$,

where $x$ is CIR and $\phi(t)$ is deterministic and chosen to fit the initial term structure.

My question is if it would make sense to consider a displaced CIR++, that is that $r(t)+\alpha=x(t)+\phi(t)$. My immediate thought is that the $\alpha$ does not provide any additional value for the model, and that the $\phi$-function already makes it possible for the short-rate to be negative?

You are right. In the CIR++, $\alpha$ parameter is absorbed into $\phi$. With the CIR++, $\phi(t)$ will allow you to have to have negative rates. You will calibrate your $\phi$ to fit the discount factors.
A simple shifting trick is to put $r(t)-f$ instead of $r(t)$ under the square root in your expression. Then $f$ is the new, possibly negative, interest rate floor. If, for example, $f=-100bp$ then the process is defined for all $r(t)>-100bp$.