How do I take all enrolled as well as unenrolled contacts from a specific sequence to have them go to a new step which is a 'Sequence' itself?

Within Uptics, you do have the ability to enroll prospects from one sequence to another sequence by just adding a 'Sequence' step within an existing sequence/campaign. Refer to this article.

If your prospects are enrolled via a smartList or manually in the existing sequence, even though they have been unenrolled manually, they are yet a part of the SmartList/existing sequence and will thus be auto-enrolled in the new sequence step.

But, if contacts in the existing sequence have been unenrolled due to a sequence setting criteria for example: 'Unenroll if Replies to this Sequence' in this scenario, these unenrolled contacts won't enroll in the new sequence step. You will, however, be able to run a filter from the 'Contacts' page using the "Sequence is SEQUENCE NAME" filter and "Sequences Completed is less than 1".