The oil cooler (from memory it's been a while since I worked on one) is held on by a central hollow bolt. It has an O ring that sits just below the nut and on top of a washer, there's another large O ring that sits a rebate at the bottom of the cooler body that attaches to the engine casing.
These O rings get brittle over time and leak, often you get oil mist first as a warning but sometimes they just start leaking, it's possible you've nudged it ot the coolant pipes while removing the oil filter. Providing it's not leaking anywhere on the body it will more than likely be the base O ring. They can suffer impact damage from stones (rare) or corrode from the inside out fairly common due to owners running the bike with incorrect coolant mix or none at all i.e. just water!
You'll have to either drain down or cap off the coolant fed pipes to remove the oil cooler. Place a block of wood or something hard flat about 50mm thick (at most) under the centre stand and place the bike up in the stand as normal. With the bike flat and
secure remember the bike will be higher it
MUST be secure, tip the bike backwards onto it's rear wheel. A car jack under the headers works a treat, or tie downs straps through and under the top yoke up to the garage beam or such like. You only lift it so as the tyre is touching the floor
DONT lift the bike, this will tip the oil back away from the oil cooler and stop leakage when the central bolt is removed. If you can't do that you'll have to drop the oil out and put it back in (as it's new) once the oil cooler O rings are replaced.
Lets us know how you get on