A player who is “as cool as the other side of a pillow” is someone who is calm under pressure. Stuart Scott (1965-2015), an anchor on ESPN’s SportsCenter, popularized the catchphrase in the early 2000s, but did not coin it.
 
“As cool as the other side of a pillow” has been cited in print since at least 1948 and has been used in sports since at least 1963.
 
   
Wikipedia: Stuart Scott
Stuart Orlando Scott (July 19, 1965 – January 4, 2015) was an American sportscaster and anchor on ESPN, most notably on the network’s SportsCenter. Scott regularly covered the NBA and NFL for ESPN. He was well-known for his unique style and use of catchphrases.
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Catchphrases
He became well known for his use of unique catch phrases, following in the SportsCenter tradition begun by Dan Patrick and Keith Olbermann. The catchphrases included:
 
. “Boo-Yah!”
. “Hallah”
. “As cool as the other side of the pillow”
   
Google Books
The Reader’s Digest
Volume 53
1948
Pg. 47:
As cool as the other side of a pillow.
 
3 August 1950, Winnipeg (Manitoba) Free Press, pg. 5 ad:
Garments as cool as the other side of a pillow.
(Eaton’s.—ed.)
   
Google News Archive
11 March 1963, The Daily Reporter (Spencer,IA), “Cousin Cuz: That Last Shot Was Inevitable” by Cuz Ottavi, pg. 3, col. 1:
The kids looked as cool as the other side of a pillow, and their first half performance showed it as they sported a six-point halftime lead.
 
Google News Archive
17 February 1964, Daytona Beach (FL) Morning Journal, “Auto Racing: Junior Won’t Change Style,” pg. 13, col. 1:
Johnson, who outwardly gives the appearance of being as cool as the other side of a pillow, is now in his 10th year of NASCAR racing.
 
Google News Archive
17 April 1982, St, Petersburg (FL) Independent, “Heresy at the EPA: Administrator does what she’s supposed to do” by James J. Kilpatrick, pg. 9-A, col. 1:
She is given to slow smiles and soft speech: she is as neatly composed as a measure by Mozart, and under the unrelenting fire she remains as cool as the other side of a pillow.
   
Google News Archive
22 September 1986, Lewiston (ME) Journal, “Patriots are shocked” by Ron Borges (Boston Globe), pg. 1B, cols. 3-4:
They were as cool as the other side of the pillow, especially after Stanley Morgan applied what appeared to be a knockout blow with a 30-yard touchdown catch midway through the fourth quarter that was his second of the period, third of the game and uplifting enough to make the score 31-21 New England.
 
Google News Archive
2 June 2002, Beaver County (PA) Times, “Boo-yah: ESPN’s Scott back from injury” (Knight-Ridder Newspapers), pg. E7, col. 1:
Next week he’ll cover the NBA Finals and seem—to use another of his catchphrases—“as cool as the other side of the pillow.”
 
Urban Dictionary
cool as the other side of the pillow
A state of being very calm and collected, rarely freaking out or becoming over-emotional, and generally well-respected by the majority of people. May have originated from ESPN’s SportsCenter.
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by Nick D October 20, 2003
   
Heavy
Stuart Scott: 5 Best Catchphrases
Published 10:00 am EST, January 4, 2015 Updated 12:42 pm EST, January 4, 2015
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2. As Cool as the Other Side of the Pillow
Likely Scott’s most memorable catchphrase, he used this to describe a player who had a career game. In this context, he’s saying whatever the player does is done right and with ease.
     
Los Angeles (CA) Times
Stuart Scott dies at 49; ESPN sportscaster with catchphrases
By STEVE CHAWKINS
JANUARY 4, 2015 6:29 PM
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Scott, who praised unflappable athletes for being “as cool as the other side of the pillow,” said he never strained for clever descriptions.