As Jen said you can create the poop pocket in the covers to help with the poop on covers.
Another option is to use a snappi with a jelly roll. Using the jelly roll creates gussets to help contain the poop.
What kinds of covers are you using? Bummis Super Whisper Wraps are hard to clean up once they've been pooped on. In our store, most families with Newborns/and or breastfed infant poop prefer the Thirsties Diaper Covers or the Bummis Super Brite Covers. These covers have the laminated (slick part) on the inside and can be easily wiped down with a wash cloth or run under the sink for easy clean up. Also, they have very little staining (except on the edges).
Now that we're having some beautiful weather here in Boston, after you've washed your covers, put them to dry in the sun. On your porch, outside, even just in a sunny spot near the window in your home will work wonders for the stains. It will work wonders on the stains and the sun is a natural disinfectant.
Feel free to stop by Diaper Lab anytime, and Laura or I can give you a hands on tutorial with your little one.
201-A Highland Avenue
Somerville, MA 02143