Yes, and the mechanism that makes this happen is called a detainer.
When another state has an outstanding warrant or unresolved legal matter involving you, they file a detainer with the facility where you are currently serving time. That detainer is essentially a formal notification to the releasing facility that another jurisdiction has a legal claim on you and wants to take custody when your current sentence ends.
The process is generally well coordinated. The state holding the detainer has access to your projected release date and plans the transfer accordingly. In many cases their transport officers arrive on or around your actual release date, sometimes waiting in the parking lot before the paperwork is finalized on the releasing end.
You do not walk free in the gap between finishing your sentence and the other state arriving. The detainer prevents release into the community and you are held in the facility's admissions area until the transfer is completed.
What happens after that depends entirely on what the other state's case involves. If it is a warrant for a minor matter, resolution may come quickly through a court appearance and possible release on bond. If it is a serious charge or a parole violation, you could be looking at additional custody in that state while the matter works through the courts.
If you or someone you know is approaching a release date with an active detainer from another state, getting an attorney familiar with interstate detainer law involved early is the smartest move. They can communicate with both jurisdictions and in some cases negotiate a resolution before the transfer even happens.