This is why most trans people learn to take what their doctors say with a huge grain of salt. "Trust your doctor" is shit advice when it comes to trans medicine. It's not just bad advice, it's Stockholm Syndrome.
I'm assuming you're on weekly shots? If so, the 4 week guideline comes from hospital settings. In hospitals, they'll have vials of medication shared between many patients. They're discarded after a month. But those vials aren't being used for a single injection once per week. They're often being used many times by many different patients. It makes sense to be more cautious due to the number of times the stopper is being pierced over a short period of time.
But for individual use? It's completely unnecessary. As long as your vial isn't visibly cloudy, has visible debris in it, etc., as long as everything is fine? Use it til it's empty.
As far as what to tell the doctor, simply don't tell him. If he asks, lie and say you threw it out after a month. Again, "trust your doctor" is shit advice in trans medicine. Your doc just learned in his training "throw out vials after a month" and never bothered looking in to where those rules came from.
Instead, use this as a means of building up a stockpile. Yes, there's no actual shortage of estrogen, but if you're in the US, do not assume your access to E is safe. All it would take is one illegal executive order stating that "it's against FDA policy to prescribe trans HRT," and all the insurance companies would stop covering it immediately, complying in advance with the illegal directive. Same thing with big hospitals and doctor networks. In 2026, it is nothing less a form a self-harm to be a trans person not stockpiling when you have the option to do so.
If unopened, those vials will last a decade if kept in a cool dry place. You can use what you're observing here to build a stockpile. Keep using the vial until it runs out. But actually fill your prescription every four injections as your doctor is prescribing.
Again, do not simply rely on your doctor. If trans care is severely affected, he'll simply stop seeing trans patients and be otherwise unaffected. You will be left high and dry without life-saving medication. Keep stockpiling until you have a decade or so of vials saved up. Always draw from the oldest vials first. Once you reach the maximum practical stockpile size, either only fill the rx every few months, or fill the rx monthly and give away your excess vials to local trans people in need.