If you had insurance for any portion of that three-month period, then the penalty would be waived.
The IRS says that if you had insurance for even one day of the month, then you can report you had insurance for the month. they also say that you can get an exemption for a short-term gap of up to two months.
If you had no insurance for three months, then for 2016, you would not be eligible for an exemption because of your change in employment.
You might want to check the federal healthcare web site to see if you qualify for this or other exemptions. They have a simple screening questionnaire that you might find helpful
Can I call IRS directly and see what they can do? Or health provider? Could you please provide me with their contact information. I didn't expect the penalty would be that high.