First some basics.
Simply put, it takes equal quantities of alpha globin and beta globin to produce hemoglobin. Beta globin is produced by the actions of the beta globin gene on chromosome 11. There is one beta gene on each of the two chromosomes. Beta trait or minor is when one of these genes is deleted or mutated. Alpha globin is produced by the alpha globin genes. There are 4 alpha globin genes, 2 each on the two chromosome 16. When there is a two gene deletion, in most cases, both genes are deleted from one chromosome. This is called in cis. When one alpha gene is deleted from the two different chromosomes, it is also a two gene deletion, but called in trans. Two gene deletions are alpha minor, as opposed to the single gene silent carrier state.
It sounds like your daughter is a two gene affected alpha minor. This also means that one of her parents carries the same thing. With the reduction in alpha globin that is produced, it often means that the patient will have anemia that is not related to low iron. Because of this, if iron deficiency is also suspected at anytime, it would require a full iron panel of tests to confirm before giving her iron. Iron should not be given to thal minors unless deficiency has been demonstrated. What may help some would be for her to take folate and sublingual B-12 supplements to help meet the higher demand for these nutrients in building red blood cells, RBC's. Avoidance of chemical fumes and artificial scents will often also help the patients, as exposure can sometimes cause the more fragile RBC's to break down. Broad beans like fava beans and medications containing sulfur should also be avoided. A diet with plenty of antioxidant foods is recommended. Many minors find an afternoon nap can help them make it through the day.