For a quick fix you could get that smaller hole wire and cut a larger piece than the existing chain link fence, (running the smaller hole wire past the chain link), and you could attach it using those plastic wire ties or zip ties to cover the gaps. The piece that goes between moving portion may not last long-term, though, because every time you go in and out it is going to bend and flex. If that is definitely what you want to use, and keep baby chicks in and rodents out, you will need smaller hole wire mesh (hardware cloth) for sure as the fencing openings are too large.
The best option would be getting different hinges so you won't have that large gap, but I do not know much about chain link fencing or if they even offer different hinges. I recall from other metal fencing I have seen over the years, there is always a gap. The best fencing for no openings would be something you build, either with pressure treated lumber or the vinyl gates. I know you're trying to work with a budget, though. (Aren't we all!)
If you want something that lasts longer, you could get more fence post material, I don't know what they call it, chain-link fence galvanized piping? Get more pieces of that galvanized piping, 6' tall but larger dimension than what is there, to help fill the gap, but not too large so that once you attach it by nuts/bolts to the other pipe, it can still function and open as the door should. It would need to be cut in pieces where the hinges are.
Sorry I can't be of more help, just not experienced with that type of fencing.
The best option would be getting different hinges so you won't have that large gap, but I do not know much about chain link fencing or if they even offer different hinges. I recall from other metal fencing I have seen over the years, there is always a gap. The best fencing for no openings would be something you build, either with pressure treated lumber or the vinyl gates. I know you're trying to work with a budget, though. (Aren't we all!)
If you want something that lasts longer, you could get more fence post material, I don't know what they call it, chain-link fence galvanized piping? Get more pieces of that galvanized piping, 6' tall but larger dimension than what is there, to help fill the gap, but not too large so that once you attach it by nuts/bolts to the other pipe, it can still function and open as the door should. It would need to be cut in pieces where the hinges are.
Sorry I can't be of more help, just not experienced with that type of fencing.