Spanning tree will make sure that all traffic from one switch to another switch only has a single path, and that is through the root switch, which really needs to be the distribution switch. The best that you could hope for is that your dashed red lines end up being redundant paths that kick in only when the link to the distribution switch fails.
STP guarantees a loop-free path. The way it does that is to create a single path that is most direct to the root switch. All other paths are placed in blocking. If you had multiple paths that allowed layer-2 loops, then your network would crash with broadcast storms.
A best practice is to only allow a VLAN on a single switch. You can have multiple VLANs on a switch, but they should not extend to other access switches. The helps eliminate the possibility of STP problems. It is no longer necessary to have everything connected to a layer-2 network. We live in a layer-3 world, and routing has become cheap, and it introduces security points and eliminates STP problems. It used to be, "Switch where you can, route where you must," but that is no longer the case.