Thyroid Blood Test Results - Any Other Test Needed for Evaluation?

Hello,
I recently had TSH level and T4, Free level blood tests and the results are below:
TSH 4.79
T4, Free 1.2
My TSH in January 2025 was 3.99. I had fluctuating TSH levels about 10 years ago with no treatment as I had no symptoms. I am now having hair thinning, fatigue some days, and have put on 8-9 lbs in the last 3 months. I am 65 years old and in otherwise good health. My PCP is talking about putting me on levothyroxine 25 mcg to correct the elevated thyroid-stimulating hormone level. Before I agree to this RX, are there other blood tests that should be done? Should I see an endocrinologist for a deeper evaluation? Or, should I pause and just have my TSH and T4, Free checked again in 3 months? Your insight would be appreciated. Thank you in advance, Jamie