Although what you are saying is possible, it is bad design.
If you are doing looped connections on your access layer switches you will always have a blocked by STP link.
If you have 2 separate closets in the new building you should endeavor to have both closets connect to both core switches with an uplink port-channel (aggregate).
Are your core switches a stack (logical or physical) or are they two separate entities?
Are those core switches in a separate location?
If they are in a separate location and you only have a fiber bundle between core 1 location and closet 1 in your new building and conversely only have 1 fiberbundle from core 2 to closet 2 and fiber bundle between both closets you should consider using a fiber pair loopthrough from one closet to another so they can still be logically connected to both core switches.