Couple of things to consider and you can take it as advice and do what is best for you -
1. I am guessing (since all most all do this) that this 3 months is full complete 3 months of calendar months and your insurance starts from 4th month day 1. If you have some flexibility, start new job at the very end of month or on day 1 of the month.
2. If you have some PTO/Vacation left in your old job and your manager can work with you, give notice in advance, and make last day of work as such that you can have one full month of insurance from old company. This way you will only have to buy 2 months worth of insurance or be without it for 2 months. For Example, Lets say you want to start new job from August 1st (or End of July like 29, 30 or 31st). In your old job talk to your manager and say that your last day of job will be August 2nd but you will use your vacation for that week - and drop your computer on Friday only. Communicate well with your Manager in advance. Or take off Friday on new job and go to old job for last day. This way you will have the insurance for August from old company. Insurance starts from 1st of month (unless company insurance starts from day 1 of employment) and only ends at the end of the month ( I found out this recently). And you even do not have to pay via pay check since you will have no pay check. Still holds true. I did that recently in DEC 2018.
3. From Tax filing, I had no insurance for 3 months in 2017 (same your case) and you can file the waiver during tax filing and shorter periods are waived. You do not have to pay penalty but make sure to file waiver during tax filing. Do not remember the form no/name.
Hope this helps in making your decision.