A new spin-exchange optical pumping polarized 3He target has been used to measure angular distributions of A0y , Ayy , and Axx for p+3He elastic scattering at proton energies of 2.28, 2.77, 3.15, 4.02, and 5.54 MeV. The new data, along with previous cross-section and beam analyzing power measurements were included in a global phase-shift analysis of data below 12 MeV. The additional data resolved the ambiguity in that analysis, and a unique set of energy-dependent phase-shifts was obtained. These phase-shifts, as well as the data themselves, were compared to theoretical calculations using realistic nuclear potential models that accurately reproduce NN scattering data. These calculations underpredicted A0y , in addition to the previously-observed underprediction of Ay0 known as the Ay puzzle. The spin-correlation coefficients were found to be better reproduced. There was good agreement for the S-wave phase-shifts, while the P-wave phases showed small discrepancies, especially for 3P2 and 3P0 , and different discrepancies were observed for different interaction models.