I am assuming it is a short term single entry tourist visa. Given the circumstances, I suggest you provide new offer letter with date of joining highlighted along with relieving letter from old company with date of last working date.
Though this does not work as a replacement for NOC, the documents tells that you are not intending to move as an immigrant. Please note that the country to which you are applying and your travel plan also matters. If the country you are visiting is well known for tourism and your stay arrangements are within the tourist places, your application might be considered.
I have a friend who changed 3 jobs and travelled to three countries (including 1 within schengen area) in between her jobs. So I know they will not turn down your application outright because you don't have an NOC.