Aluminum-doped, oxygen-deficient YBa2Cu3-xAlxO6+delta single crystals with different Al contents x (0 less than or equal to x less than or equal to 0.19) and O contents (0.18 less than or equal to delta less than or equal to 0.36) were studied by magnetic neutron diffraction. All of the Al-doped single crystals show two magnetic transitions, the first between the paramagnetic state and the AFI phase, and a second transition at low temperatures between the AFI and the AFII phase. The Neel temperature T-N of the antiferromagnetic AFI phase is found to be insensitive to the Al content x as well as the O content 6+delta in the x-delta region investigated so far. In a limited temperature interval the order parameter shows the components of both the AFI and AFII phases indicating competing interactions. For some crystals a complete reordering to the AFII phase at 4.2 K can be observed. Although the antiferromagnetic ordering pattern is different for the AFI and AFII phase, the ordered moments on the Cu sites are within the experimental error ([S](Cu(2)) approximate to 0.56 mu(B), [S](Cu(1)) approximate to 0 mu(B)) identical in the two phases. Comparison of Al-doped crystals with pure crystals of the same oxygen content shows that the presence of excess oxygen only (delta>0) is not sufficient to obtain the AFII ordering at low temperatures. Instead, the AFII ordering is triggered by the creation of the local magnetic moments in Cu-O-Al chain fragments.