Do you free range your chickens at all? If so, try to stay with safe plants.
I agree with Killer3Bs, vining things will work nicely.
I planted tall peas by my run each year (4-5 feet tall). The pea plants are edible, so if they grew through the run wire, it gave them something to nibble on, plus I would snack on the peas when I went to care for the chicks

. This would be great while you are waiting for something more substantial to grow. They can also be interplanted with nasturtiums and or marigolds as
@Alaskan mentioned. (both also completely edible) Might also be able to get a harvest in in early spring before a deciduous tree fully leafs out once it is established.
Again,
@3KillerBs is right on the money.
Semi-Dwarf apples or
pears, or even
crab apples (not the ornamental ones, as they don't bear fruit, or it is tiny if they do). You could also plant a
ChokeCherry. Depending on the particular variety, it is a shrub to small tree. It is native, and wildlife like it, too. If you free range, this could become a favorite roosting spot in addition to bearing 'treats' in the fall! If you don't want something quite as tall, then maybe
highbush blueberries(native), or any kind of commercial variety of blueberries.
Below is a link to the Mass Wildlife's site for native shrubs for wildlife food. All should work for you in CT, too. Sort/search by soil type. Just beware of ones that have thorns or prickers (Hawthorn, for example) if you are going to plant it where that may become a 'pain'
https://www.mass.gov/guides/native-shrubs-for-plantings-as-wildlife-food
I don't know the orientation of your coop, so keep this in mind. If possible, a tallish shrub/shorter tree planted to the south-west of the coop/run would give best shade at the hottest time of the day.
A shorter shrub/bush or tall grasses like:
on the east/southeast would help with early morning sun.
A tall tree (I'm talking 20 or more feet tall) planted 8-10' (or more depending on the tree) to the South will help once it gets large. (May have to keep branches pruned so don't rub against run when it gets to intermediate stage.... but as gets large, outer branches will overhang run, and provide shade to the south, creating a slightly cooler micro-climate in the run area. Depending on the size of the tree when purchased & it's growth habit (i.e. fast versus slow), though, this could be a 8-10 year venture to get it to canopy size. Pay attention to tree shape when mature...you want to buy something that is more open/umbrella shaped

than pyramidal shaped

when mature.
I know you didn't ask this, but you might want to plant an evergreen to the North (or direction of prevailing wind) to provide a wind break in the winter when it is cold. That, combined with a deciduous planting to the South (South, south-west, South-east) would make for 4 seasons of more comfortable living
I would look over your options, make a plan, and then slowly add plants over time as you can manage (it can be a HUGE task trying to do it all at once if you are going to go whole hog with landscaping around the coop/run) Anything you do will be better, and your chooks



will appreciate it.
Hope this helps!