The second proviso seems to empower the C.J.M. or the Sessions Judge of the district in which the arrest is made to release the person (against whom the warrant is issued) on bail even if the offence is non-bailable on consideration of accompanying information and documents.
This is completely a new proviso inserted in the Code with a view to mitigating the hardship involved in taking the person to the Court which had issued the warrant. But the Delhi High Court in Arun Kumar Singh v. State of Delhi has held that the second proviso to Section 81 does not envisage the grant of bail by the C.J.M. or the Court of Session having jurisdiction over the place of person arrested by the police without warrant.
The reason being that in the context of Sections 78 (2) and 81, the law has not given such enabling power to grant bail to Courts other than one having jurisdiction over the area of the commission of the offence.