Skaggs, who will miss 10-12 weeks with a Grade 2 right oblique strain, joins ace right-hander Garrett Richards and top relievers Cam Bedrosian, Huston Street and Andrew Bailey.
In 2016, Richards and left-hander Andrew Heaney suffered season-ending elbow injuries within the first month of the campaign. Their losses exposed the Angels' thin pitching depth and played a significant role in derailing the rest of their season.
Though the Angels are experiencing a similar stretch this year, Scioscia said he believes the club has a deeper pitching staff that is better equipped to withstand the current injury wave. The Angels have used 20 different pitches in 2017, the most in the Majors, yet finished April with a 14-13 record.
"We went through a similar run last year. It obviously impacted a lot of the things we hoped to do," Scioscia said. "I think coming out of spring, we're a little deeper this year, which helps. We've seen some guys come up and keep us in games.
"We hope that some of the guys that are banged up, it's not as devastating as last year when we had [torn] ulnar collateral ligaments and Tommy John surgery and things like that. If these guys can come back and pitch for us, it's going to be important."
Still, Scioscia acknowledged that the news of Skaggs' diagnosis was disappointing. Skaggs struggled in his first two starts of the regular season but had recently emerged as perhaps the Angels' best pitcher, allowing just three runs over 19 innings in his last three outings. The 25-year-old held the Rangers to one run over five innings on Friday but was forced to exit after feeling a "pinch" in his right side.
The Angels will have to fill Skaggs' spot beginning on Thursday, and Scioscia said the club is still weighing its options. Daniel Wright, Alex Meyer and Vicente Campos are among the candidates to slide into the rotation, as is Yusmeiro Petit, who is currently in the Angels' bullpen as the long man.
"I think we have guys that are going to go there and keep us in games," Scioscia said. "We'll have to make a decision on Thursday, but I think for the most part, some guys are getting more to their length. That's going to help. We've been challenged a little bit with some setbacks, but we'll keep going."