Recent low-temperature heat capacity (CP) measurements on polycrystalline samples of the pyrochlore antiferromagnet Tb2+xTi2−xO7+δ have shown a strong sensitivity to the precise Tb concentration x, with a large anomaly exhibited for x∼0.005 at TC∼0.5 K and no such anomaly and corresponding phase transition for x≤0. We have grown single-crystal samples of Tb2+xTi2−xO7+δ, with approximate composition x=−0.001,+0.0042, and +0.0147, where the x=0.0042 single crystal exhibits a large CP anomaly at TC=0.45 K, but neither the x=−0.001 nor the x=+0.0147 single crystals display any such anomaly. We present new time-of-flight neutron scattering measurements on the x=−0.001 and the x=+0.0147 samples which show strong 12,12,12 quasi-Bragg peaks at low temperatures characteristic of short-range antiferromagnetic spin ice (AFSI) order at zero magnetic field but only under field-cooled conditions, as was previously observed in our x=0.0042 single crystal. Furthermore, the frozen AFSI state displays a gapped spin excitation spectrum around 12,12,12, with a gap of ∼0.1 meV, again similar to previous observations on the x=0.0042 single crystal. These results show that the strong 12,12,12 quasi-Bragg peaks and gapped AFSI state at low temperatures under field-cooled conditions are robust features of Tb2Ti2O7, and are not correlated with the presence or absence of the CP anomaly and phase transition at low temperatures. Further, these results show that the ordered state giving rise to the CP anomaly is confined to 0≤x≤0.01 for Tb2+xTi2−xO7+δ, and is not obviously connected with conventional order of magnetic dipole degrees of freedom.