Your levels are on the low end of normal; you are not anemic. However, there are some researchers such as those at the Cleveland Clinic who believe that a level of 70 ng/mL is needed by hair loss patients to maximize hair growth. One example: http://www.foxnews.com/story/0,2933,196057,00.html
The research is not absolutely conclusive, but it would appear that having below-average levels of iron can be a contributing factor to hair loss and that increased iron levels can promote recovery, even for those patients who aren't anemic.
I am not a doctor, but based upon the above, I would suggest that if your doctor approves that you take iron supplements for several months until your iron levels reach the 70 ng/mL level discussed above. (Once you've reached those levels, you should talk to your doctor about discontinuing the iron; too much iron is also bad for your health, so you won't want to take iron supplements indefinitely.)
Take the supplement on an empty stomach, with a full glass of water, and refrain from eating or drinking for an hour or so once you've taken it. Along with the iron, add separate supplements of the amino acid lysine and vitamin C, as these will improve your absorption of the non-heme form of iron that is contained in the supplement.
(Choose your supplements carefully -- the FDA does not regulate the labels, so the promises can be misleading hype, and not all brands are equally reliable. As a rule, the cheap brands are cheap for a reason; it pays to do your research.)
There are also foods that help or inhibit iron absorption, even if they contain iron in them. For example, spinach has a lot of iron, but other components in spinach prevent your body from absorbing it, making its iron content effectively near-useless for your purposes.
Iron from meat (heme) is readily absorbed by the body. Iron from plants (non heme) is not as well absorbed. Non heme iron sources should be combined with foods that facilitate absorption in order to maximize the amount of iron that you obtain from your food. (Do not assume that the iron content in food equals the amount of it that the body will absorb.) That is especially true if you are vegetarian, as the richest sources of iron are animal organs, red meat and certain shellfish (mussels, oysters, clams), not plants.
Foods with tannins (e.g. tea, coffee, red wine) and phytates (whole grains, soy, legumes, etc.) are iron inhibitors. Foods with amino acids and Vitamin C are iron facilitators. To maximize iron absorption of non-heme iron fron your food, you should combine it with a facilitating food, i.e. include a citrus fruit along with your cereal so that the iron in the grains is better absorbed as a result of the vitamin C derived from the fruit.
This does not mean that you should avoid all of the inhibitor foods -- you need grains and beans in your diet, for example -- but you need to eat them in moderation, combine them appropriately and otherwise give your body the opportunity to maximize its absorption of iron while your hair recovers. (I would, however, go easy on the tea, coffee, red wine, and soy until your iron levels have normalized.)
If you have a vegetarian diet that isn't well balanced and that includes a lot of inhibitors, such as soy, that could contribute to your iron levels being as low as they are.
Also be aware that excessive zinc and Vitamin A can also be problematic. Excessive zinc can inhibit iron absorption. A lot of multivitamins contain levels of vitamin A that approach overload, so be sure not to overdo it.
You find a lot of talk online about the value of biotin. As biotin deficiency is rare, it seems unlikely to me that it would help, and it isn't clear to me that there is much research to support the benefits of adding a biotin supplement. However, like Vitamin C, it is a water soluble vitamin, so whatever you consume that you don't need will be processed out of your body safely, unlike Vitamin A, zinc and iron, which can be toxic if you ingest too much. If you don't mind spending a few more dollars, you may want to take it as a sort of insurance, even if it doesn't end up doing you much good.
You can learn more here:
Do keep in mind that iron is not a miracle cure, and it won't help everyone. Those who have high levels of iron in their blood should avoid taking supplements, as excessive iron can also be toxic. Iron definitely isn't going to fix androgenic alopecia. Do your own homework, and be sure that you get your nutritional information from legitimate sources such as the NIH, as there is a lot of quackery and bad information online. Verify whatever you read, including this post.
Best of luck.