The roof on the house is a piece of fence so I added a tarp. Then I added a second tarp; still gets soaked. The run has two tarps and at a point they overlap.
Think you just need a larger tarp that covers it all, and hangs over each side about a foot, so if it is raining at an angle it won't just rain in.
Our runs have tarp covers and it hangs over about a foot on one side, and reaches to the ground on the other side ... that way it can be rolled up and bungie corded when nice, and let down in winter to keep north winds off chickens in run.