I have struggled with this hard in the past.
So here is what I do now...
1. The BIGGEST thing I can suggest is to avoid resizing at all costs. Say the horse is 1000x1000 but your BG is 1000x850. Do NOT resize the horse, resize the bg.
You can also blur the BG ever so slightly to make the horse stand out better.
2. Go ahead and adjust the canvas size to the horse. Say the horse image is 1000x1800. Go and find your canvas size and change it to be an exact square so anything like these: 1000x1000, 1800x1800, 1400x1400
So when you do resize it as a horse av, you don't have to worry about losing even more quality. Do the same here for banners, adjust the canvas size to anything with the same proportions as 669x150
3. A lot of the time, images can be 3000x2800 or any kind of large file like that. So here is what I do, I resize the image down after I cut out the horse and I only resize it down to around 1000x1000
and then after finishing my piece, I resize to HEE dimensions.
4. Drawing highlights and shadows on the horse really help
5. I have also found that a bright blue or fire orange eye can really pull a piece together.
6. Darker pieces hold quality easier. Lots of contrast also helps, I always adjust my pieces slightly darker with a bit more contrast before resizing.