This is in part why some people go to clinics. You are correct that most specialists are not going to send you home with a vial of test, but this is something that can be negotiated with the doctor and typically they end up allowing this to occur with a healthy track record of legitimate follow up labs. There is a trust factor that gets established as it is way too easy to abuse.
When you visit the specialist, keeping in mind that an endo is really the correct route but a urologist is good, you want full hormone labs done with a CBC/lipids. For hormone, you need:
dhea-s
estradiol (if possible, they generally won't do this at a urologist)
PSA
pregnenolone
total and free testosterone, you already got this, but a second check is always good while they are at the other tests
SHBG
thyroid- TSH, free T3, free T4
cortisol
Also, be open to discussion for options. I would elect to seek ways to naturally raise test levels to address symptoms, possibly clomid, and lastly exogenous test whether it be patches, gels, creams, or injection.