However, the Shropshire-based side have wasted no time in securing a replacement, with Askey deciding to leave National League champions Macclesfield Town to sign a three-year deal.
The 53-year-old told the club's official website: "My hopes are to get Shrewsbury promoted. Whether that will be next season, or in another year or two, you've got to aim high.
"It's all happened so quickly, but it's a great move for me. It's a big wrench to leave Macclesfield and it had to be something special for me to come. I'm really looking forward to the challenge and testing myself."
Askey has been with Macclesfield for the past five seasons, with 120 wins and 58 draws coming from his 267 matches in charge.