  We continue our study of the Thresholding Greedy Algorithm when we restrict
the vectors involved in our approximations so that they either are supported on
intervals of $\mathbb N$ or have constant coefficients. We introduce and
characterize what we call consecutive greedy bases and provide new
characterizations of almost greedy and squeeze symmetric Schauder bases.
Moreover, we investigate some cases involving greedy-like properties with
constant 1 and study the related notion of Property (A,$\tau$).
