If you would like to add the previous service to the present employment you should resign the previous employment properly and it can be a technical resignation also.
Give them a written request, addressed to a specific person, with cc's, in that case, it becomes a personal liability. Make sure the request letter is sent at least a week before the interview. Nobody should deny you the NOC. In case it is denied, you can carry a copy of request letter to your prospective employer, and submit it as a proof.
Denial to grant NOC is not sustainable as it is not the case of Ministry of Defence that there is a shortage of officers and the applicant's departure would be greatly inconvenient...Neither public interest is pleaded nor could it be established," a bench of CAT, comprising Chairman V K Bali and Vice Chairman L K Joshi, said.
There are many more such judgments to support/assist your case.
In many cases if the organization doesn't send any refusal within 30 days of the closing date of application it is presumed that your letter is accepted and no denial is made. You don't have to take any NOC during interview and the only reason why you are submitting the intimation is that, it will enable your office to relieve you on time if you are selected.