Round 16, #474 overall - Brendon Hayden, 1B
So, the Phillies want to do this one again. That's right, Hayden is a rare re-draft re-draft, as the Phillies had him in 2011 in the 36th round. Or, at least we should hope they want to do this again, now that he's not a tough high school sign. Hayden is 6'6", 230 pounds, bats left and throws right. He was a 2015 All-ACC First Team selection at first base, and was born in Wisconsin.
Round 17, #504 overall - Kenneth Koplove, RHP
After going somewhat local with an Illinois-born UPenn product in the 14th round (Austin Bossart), the Phillies get really local with this pick. Koplove attended Penn Charter before moving on to Duke University. If the name looks familiar, there's good reason: he's the brother of former MLBer Mike Koplove. Kenny was Duke's shortstop for 33 games, but also their primary closer. Unsurprisingly, the Phillies have selected him as pitcher. He struck out 37 and walked 12 in his 25 1/3 innings in 2015. And yes, he is a sidewinder, much like his brother was. (This is older video from about 3 years ago, appears to be from his Penn Charter days.)
Round 18, #534 overall - Gregory Brodzinski, C
Brodzinski attended the small, private Catholic school of Barry near Miami. The Barry website has (incorrectly) listed his hometown as "Turnesville, NJ". We're going to go ahead and assume that means Turnersville, making Brodzinski another local product. (He attended Bishop Eustace HS.) Regardless of misspellings, Brodzinski is another catcher, and another big guy, at 6'2", 225 pounds. He hit .327/.367/.469 this season. A relatively detailed (but old) June 2010 scouting report summed Brodzinski up this way:
Round 19, #564 overall - Robert Tasin, RHP
Tasin is a 6'1" right-hander from the University of Oklahoma. As a redshirt senior, he is a bit of an older prospect at this point. He appeared in 18 games this season, starting 11 of them, and went 9-2 with a 2.52 ERA. He doesn't appear to be a big strikeout guy, with just 57 in his 78.2 innings pitched.
Round 20, #594 overall - William Stewart, LHP
The Phillies finally go back to high school with this pick, taking Stewart, a Hazel Green HS (AL) product. This site has Stewart profiled, and has his fastball at 88, with a top speed of 91, and your fact of the day that he apparently wants to be a wildlife veterinarian. He has signed to play at Wallace State in Alabama.