The short of it:
1) What you're describing isn't a jet anymore, it's an "airbreathing rocket" - a rocket that uses air while in the atmosphere, oxidizer when not. Whenever you're feeding oxidizer, it's running as a rocket, full stop - your compressor and intake are now out of use, and you need a very different turbine to drive your rocket hardware like the turbopump.
2) What you're describing is totally impractical for a jetliner. Liquid oxygen is hazardous, heavy, bulky, an added expense, cryogenic, and a big long laundry list of other things.
3) Airbreathing / dual-mode rockets have been looked into for quite a long time, but they're a tough issue. Regular jets only work at relatively low speeds. Getting to orbit is not about flying high, it's about flying incredibly fast - the vast majority of the needed energy is used for acceleration. Hence you're hauling a lot of extra mass (an airbreathing engine) for a proportionally little gain. If you want to operate in airbreathing modes at higher airspeeds, you either need to burn supersonic or vastly increase the air's density. The former is known as a scramjet - an active topic of research. Examples of the latter include LACE wherein the air is fully liquefied, and SABRE (proposed for Skylon) which combines cooling with high pressures. All of them present challenges. Scramjets, for example, face "frozen combustion" problems, wherein the air is flowing so fast that in the time it takes for the propellant to burn, it's already long since moved out of the engine and thus not useful for propulsion.