What is the change fee for BOM-BLR segment?
For under $200 (maybe as low as $100) you can buy a BLR-BOM round trip. Fly to BLR on your original ticket as planned and buy a separate BLR-BOM round trip. Simply skipping your BOM-BLR segment will not only cause your reservation to be cancelled, you will still need to buy one way BOM-BLR. Travelling to BLR as originally ticketeed and buying a separate BLR-OM round trip is the best solution.

If under the fare rules, your emergency, is a qualifying emergency that allows you to change your outbound ticket, it might require you to pay "new" fare available at the time you make the change even if there were no change fee.
You have already talked to a Jet representative once. Calling again can't hurt. Depending on your status with the airline and the mood of a supervisor, you might be able to make free change without paying a higher fare. A supervisor could override any fees.
I had some emergency come up after I had bought my ticket (via Boston), that required me to stay in Boston for an unspecified time. The airline allowed me to not only stop in Boston for as long as I needed to, but also left the onward segments open. You have to e very lucky to get that. I have also been able to change my outbound due to post surgical care. The surgery itself took place more than a month after the ticket was booked but three weeks before the outbound. The first thing you should do is read your fare rules carefully to determine what you are or aren't allowed.