Close reading is slow reading. Robert Frost said it best, when he talked about being an "ear reader." "The ear is the only true writer and the only true reader. I've known people who could read without hearing the sentence sounds and they were the fastest readers. Eye readers we call them. They get the meaning by glances. But they are bad readers because they miss the best part of what a good writer puts into his work. 
There are many circumstances in life where fast reading is called for. Taking the Common Core assessments is definitely one of them. The passages and questions are deliberately designed to have you read them carefully. Students need explicit guided practice in the appropriate reading pace. When I work with teachers, I give them the passages from the New York State English Regents Exam (Common Core). To make the point that careful reading is (relatively) slow reading (reading by activiing your read-aloud voice in your head), I stop them every 60 seconds. Everyone should be at the same place in the text. 
Let your students learn what it feels like to be an ear reader. Experiment with the comprehension questions to see which method works better, "ear reading" or skimming/scanning to catch the answers. 
"Ear reading" is the best strategy I know of to foster close reading.