Strongly overdoped HgBa2Ca2Cu3O8+δ (Hg-1223) samples with superconducting transitions as low as 97 K were successfully obtained by using HgO, Ca2CuO3, CuO, and mixtures of highly oxidized BaCuO2+∈ (∈ ≈0.13) and/or BaO2 powders as starting materials for the high-pressure synthesis at 5 CPa and 950°C. The overdoped state was confirmed by observing negative values for the Seebeck coefficient throughout the temperature range from Tc to 320 K in a thermoelectric power measurement. Also, both of the cell parameters a and c were found to decrease with decreasing Tc, reflecting, respectively, an increase in hole concentration in the CuO2 planes and the incorporation of oxygen into the HgOδ layer. The oxygen excess δ as determined by the Cu(+I)/Cu(+II) coulometric titration method, was ∼0.19 in the overdoped sample with Tc=107 K. Subsequent reducing annealing in an Ar atmosphere at 280°C increased the Tc to 131 K. Consistently, only positive Seebeck coefficient values were observed up to 320 K for the Ar-annealed sample. A clearly underdoped material with Tc=118 K was obtained by annealing the same sample in Ar at a higher temperature (∼400°C). Finally, high-pressure synthesis starting from less-oxidized BaCuO2+∈ (∈≈0.06) yielded a Hg-1223 material with δ≈0.10 and Tc= 132 K.