This situation is indeed tricky, but there are ways to address it. Firstly, I would suggest speaking with your HR departments again, clarifying the situation and requesting their assistance.
If this doesn't yield results, it's time to take it up with the EPFO directly. Here are the steps you can follow:
1. Visit the EPFO grievance portal
https://epfigms.gov.in/.
2. Click on 'Register Grievance'.
3. Fill in the necessary details like your PF number, name, address, etc.
4. In the 'Grievance Description' section, clearly explain your situation, stating the overlap issue and how it's affecting your new job and EPF transfer.
5. Attach any supporting documents if possible.
Once you've lodged the complaint, you will receive a registration number. Keep it safe for future reference. Usually, EPFO addresses the grievances within 15-30 days.
If the issue still doesn't get resolved, you can escalate the matter to the regional PF commissioner of your region. You can find their contact details on the EPFO site.
In case of legal implications or if you're not getting any help from the above-mentioned sources, you can consult a labor law expert or attorney. They can guide you on the legal steps that you can take.
Remember, it's crucial to maintain all the documents related to your employment and EPFO for such situations. Having a strong paper trail can significantly help in resolving such issues.
Additionally, in future transitions, try to maintain at least a day's gap between relieving from one company and joining another to avoid such overlaps.
Good luck, and I hope your issue gets resolved soon!