Is there a way to manually order documents within a document library in SharePoint 2013? What I want is an ability to grab a document and drag it and release to a different position; once I release it, items in the document library should be reordered to new positions. Is there any out-of-the-box approach to achieve this?
As @DannyEngelman correctly points out in his comment, there is no OOTB way for doing this.
Integer field as you suggest yourself is one way, but hell to maintain.
If you are working with an on-prem solution I'd rather suggest you create your own custom FieldType in which you can maintain an order - take a look at this question for guidance.