Select all layers using Ctrl/Cmnd + a, grab any layer and drag.
Second option, the one that I would use - Create a new composition, nest your existing composition in the new one and drag it into the new position, then add your additional layer.
Remember, AE is primarily designed to create shots or short sequences. 90% of my AE projects are a single shot less than 10 seconds long. Most of my sequences are no more than 2 or three shots. You create motion graphics and visual effects in shots with AE and you edit with Premiere Pro.
What is your "main program" that you edit in? If you dont mind me asking
Premiere Pro on my equipment. Avid or Final Cut Pro on about 1/4 of the folks I freelance for. Occasionally on higher end systems like Flame (Autodesk) and once in a while I'll do a project for someone else on Nuke. About 90% is Premiere Pro for most of my work year.