Sure. (Of course not your doctor, not diagnosing or treating, and all the other cautions.)
Specifically what would help is of the CBC, the hemoglobin and the MCV (mean corpuscular volume). A reticulocyte count. The total iron binding capacity (TIBC), ferritin level, and serum iron. If done hemoglobin electrophoresis and fecal occult blood test.
The overview, potentially more than you want to know …
The first question is what specific result is actually being referred to as “levels low.”
For the sake of discussion let’s assume it is the hemoglobin level.
A common cause of persistent mildly low hemoglobin (mild anemia) is, as mentioned above, ongoing chronic blood loss, such as from heavy menstrual flow, or from another source such as in the GI tract. This causes iron loss and the body needs iron to make new blood cells to replace the loss. Ongoing loss and inability to make enough new ones leads to anemia. It is not the only cause, just a very common one.
In that case the red cells floating around tend to be older ones (as there are fewer new ones being made) and older red blood cells (corpuscles) are smaller … hence there is a low mean corpuscular volume (MCV) aka “microcytic” anemia. They also have less color, IOW are “hypochromic.” If tested the reticulocyte (a step that is a precursor to new RBCs) should be low.
So if there is a low MCV (microcytic) anemia then iron deficiency often from ongoing loss is often the leading suspect. If female and after menopause probably checking stool to make sure that is not the source of ongoing loss would be prudent. If the MCV is not low then right off it is likely not due to iron deficiency and iron won’t help. A whole different differential.
But iron deficiency is not the only cause of low MCV anemia and if there is no source of loss and no significant response to iron supplementation, then it is likely reasonable to confirm that there is iron deficiency and possibly to check for other main causes of low MCV anemia. First actually checking the various measure of iron status more directly: the serum iron level (low in iron deficiency); the TIBC (should be high in iron deficiency as the body has more available sites, more capacity, to bind iron on the various carriers of it in the blood); and ferritin (which should track along with serum iron. If that pattern is not present then the diagnosis should be questioned.
Other fairly common causes of low MCV anemia include lead elevation, chronic inflammation from chronic disease, and probably most common in someone who is otherwise well, beta or alpha thalassemia trait. More iron won’t help any of those.
About the thal traits - hemoglobin is actually made up of two different kinds of proteins, called alpha and beta globins. Beta globin is the one involved in beta thal; alpha in the alpha one (aint we clever). We get one copy of the gene that controls each of their production from each parent and having two defective copies is bad but one defective copy coupled with one normal copy just causes a mild anemia if that. One in four chance of passing the gene onto a child and if the other parent also has trait a one in four chance of the child have full blown disease. Prevalence of the thal traits varies by ethnicity so possibly a more likely explanation if someone is of Mediterranean, African or Asian ancestry.
Hemoglobin electrophoresis can test for a thal trait and if that is the case then iron can be stopped (not going to help and not causing any problem) and the children of the person may want to know if they are carriers too since if they are they might want to know the status of a partner before having children together.
Hope that helps.