"shooting higher" is a correct phrase and can be used in written English.
It means aiming for a higher goal or setting higher standards. Example: "In order to achieve success, we need to keep shooting higher and never settle for less than our best effort."
Exact(13)
"We're shooting higher than that," guard Joe Johnson said.
The Yankees want the player to be a midlevel prospect, and the Twins are shooting higher.
Soon the pair almost repeated the dose, Lennon crossing and Modric shooting higher as Cech scrambled away.
"Durant is shooting higher than 50 percent for the first time in his career," Lee Jenkins writes at Sports Illustrated.
Despite 38 victories on the PGA Tour, Woods has never won a tournament after shooting higher than 72 in the opening round.
But investors and analysts applauded the results, sending bank stocks shooting higher and, in the case of the Bank of Nova Scotia and the Candian Imperial Bank of Commerce, touching new yearly highs.
Similar(47)
Stoke squandered several opportunities by shooting high or wide.
I like to think there's something of Sachs in me, but that's shooting high.
Our own columnists think Ban Bossy is either just the beginning or not shooting high enough.
I suppose people shooting high-definition video might find these useful.
Session started out shooting high-quality booty-shake videos for World Star Hip-Hop.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com