Looking for ideas on how to upgrade IOS on a fleet of ~450 devices

The situation:

• In my 2nd year as a k12 jr. neteng
• ~450 Cisco devices under my supervision, as an MSP-level netadmin, for various local school districts
• The bit of research I've done so far suggests the number of devices requiring IOS updates approaches 100%
• Almost all devices are of the 3750/3560 vintage. A few 2800 routers here and there, and yes, a few even older than those.

Side-note: I understand these are ancient devices, and the Best Solution(R) is to refresh them. This is on my radar, and I'm working with the resources I have to make this happen. For now, I need help upgrading IOS. If you have constructive advice, I'm all ears! If all you have to offer is something about how old these devices are, please feel free to not reply.

Tools I have:

• SolarWinds Orion NMS in production
• Decent python/napalm/netmiko skills—I've done more than one useful thing with them
• Very, very basic ansible skills—just did my first PoC recently, nothing serious
• A supportive manager
• A reasonable budget for more tools if needed