I'm 41, 7 months pregnant for the first time after trying for 6 years. For me it was definitely a hormone thing, but not the simple tests your medical doctor will probably do.
Defintely get your LH and FSH tested, and have the die test. Definitely ask to have your thyroid tested, not just the first level, but all three - TSH, T-3, T-4. Definitely get every test that your insurance will cover! From what others have communicated, I am so glad to know that there are progesterone therapies that are helpful in your situation.
But as far as boosting fertility, I had my thyroid checked a couple of times, and all came back normal (low end of normal, but still normal).
Then I started seeing an acupuncturist (Patricia Lollis ###-###-####) and she had me do a more comprehensive test series that checked my body's ability to produce sertonin, cortisols, adrenals and GABA functionality, all of which are tightly related to fertility, and all of which can have reduced functionality due to againg & diet. Turned out my body wasn't producing anywhere near the right amount of serotonin or adrenal hormones. Three months of strict diet and prescribed sero/adrenal supplements and I got pregnant without thinking about it.
And I also feel the best I've ever felt in my life.
So I encourage you to keep faith in yourself and your body, and ask for more testing, go to other resources if your medical doctor just does the standard round of tests.
My heart goes out to you, as miscarriage has been my biggest fear at my age. At 7 months and everything normal, I know it's possible to have a health pregnancy at my 'advanced'! age. :-)