Where are you wanting to teach? If in a language centre, it probably wouldn't be an issue - but as someone with proper teacher training you are likely to find such a job soul-destroying. If you do want to go that route, then language centres generally hire year-round.
If you are looking for a position in a *real* school, then they will want you to have not only your degree but also to have completed your supervised teaching year(s), QTS, full state certification, or whatever they call it in your country; there is very little scope for fresh graduates who need a visa.