The .22 Remington Jet was based on the .357 Magnum necked down to .22 caliber. It gave about .22 WRM ballistics in a revolver. Facotry chambered in the S&W M48, but lots of users complained of case set-back on the recoil shield, and locked up revolvers.
The .22 Improved Jet was made by firing .22 Jet in a reamed-out or "Improved" chamber, with near-straight sided case, sharp shoulder and one-caliber neck. It gave slightly better ballistics than the standard .22 Jet, but less tendency to lockup the S&W revolver it was chambered for.
As far as I know, only the S&W M48 K-frame revolver, the Thompson-Center Contender, and a few custom rifles and single shot pistols have ever been made for these rounds.
HTH
John.