The affinity of hemoglobin for O2 is increased by:

Production of red blood cells is under the control of differentiation inducers that promote differentiation of uncommitted pluripotent stem cells into committed stem cells ("colony-forming units") and growth inducers that promote maturation but not differentiation of the cells. Erythropoietin, the principal factor that stimulates red blood cell production, promotes both differentiation of stem cells to proerythroblasts and maturation. Several other factors, including androgens, also stimulate bone marrow cells and enhance production of erythrocytes resulting in higher hematocrit and hemoglobin values in the male. Blood loss during a menstrual period averages 30-60 mL and cannot explain this difference. Ref: Kennelly P.J., Rodwell V.W. (2011). Chapter 6. Proteins: Myoglobin & Hemoglobin. In D.A. Bender, K.M. Botham, P.A. Weil, P.J. Kennelly, R.K. Murray, V.W. Rodwell (Eds), Harper's Illustrated Biochemistry, 29e.