These types of jobs can be tempting, but there are pitfalls to them.
The first that comes to mind is, is the Citation III FAA certified for one pilot or two? If the FAA certified it for one pilot, then the time you spend as a First Officer on it is not logable. I don’t care what their insurance requires or what fable the charter company tells you about part 91 sole manipulator of the controls, if you are not signing for the airplane, you cannot log FO time in a single pilot jet. If the FAA certified it for two pilots, you are good to go.
The second issue is that you are not going to fly very much as an FO and the airlines know this. You will get really good at talking on the radios, getting coffee, washing jets, etc, but not flying. Most charter companies very rarely let the FO fly. So even if you can log the FO flight time, it is likely not quality FO time.
Third issue is that the jets are small, lightweight and not really reflective of airliners. A RJ pilot gets experience working in an airline environment, exactly the same as the majors. While corporate experience is valuable, it is just not the same as the airlines. If I wanted to be a ankle surgeon I would do my residency in orthopedics, not radiology.
Very, very few people get hired at the major airline published minimums. Those are for special cases and for military pilots who do not fly as much. I would recommend the regional route to you, but that is just my opinion.