Notwithstanding any provisions of law that impose a period of limitation to the contrary or any provisions of any other law pertaining to the filing of a notice of claim or a notice of intention to file a claim as a condition precedent to commencement of an action or special proceeding, with respect to all civil claims or causes of action brought by any person to recover damages for physical, psychological, or other injury or condition suffered while under the jurisdiction and in the care and custody or supervision of: the state department of corrections and community supervision, except a person under community supervision as defined in subdivision thirty-one of section two of the correction law, a hospital as defined in subdivision two of section four hundred of the correction law, a correctional facility as defined in subdivision three of section forty of the correction law, a local correctional facility as defined in subdivision two of section forty of the correction law, or an alternate correctional facility as defined in subdivision one of section eighty-seven of the correction law, the time in which such action must commence shall have the full benefit of the limitations period provided in the relevant section of law and, in addition, shall have a two-year window following the date of the person's release from the relevant period of custody within which to commence an action based on such injury or condition. For any claim that would have been time-barred but for the application of this section, no notice of claim or notice of intention to file a claim shall be required as a condition precedent to the commencement of such action or special proceeding. This exemption shall not apply to any claim that was timely under the otherwise applicable statute of limitations.