I’m assuming your switches are properly stacked, one stack for the core, and one stack for the admin switches. If so, then you might want to look at link aggregation rather than using single links and relying on STP. Link aggregation bundles the links and means you can use the aggregate bandwidth of the links, rather than relying on STP which blocks all paths except for one.
I would take four ports on your admin switch stack, say ports 1 and 2 from switch one, then port 1 from switch two, and port 1 from switch three, and aggregate them all (under Switchports in the Dashboard select the four ports then click Aggregate, you should see an AGGR port appear at the end of the switch port list). On the core switches I’d aggregate ports 1 and 2 on both switches (same process to create another AGGR port). Then you put the four cables in between the two switches and the links should come up as a single logical 4Gbps link - no need for spanning tree.
The only Spanning-Tree change I’d recommend you still make is to ensure your core switch becomes the root bridge. Go to Switch -> Switch Settings, then under STP configuration, click the ‘Set the bridge priority for another switch or stack’ link, find your core stack and the set the priority low to make it the most likely root.
Hope this helps.