The goal of this note is to study the performance of the Thresholding Greedy Algorithm (TGA) when we increase the size of greedy sums by a constant factor $\lambda\ge 1$. We introduce the so-called ($\lambda$, partially greedy) bases. While the case $\lambda = 1$ gives strong partially greedy bases, we show that, for each $\lambda > 1$, there exists a (Schauder) basis that is ($\lambda$, partially greedy) but is not strong partially greedy. Furthermore, we investigate and give examples when a basis is 1. not $1$-(almost) greedy but the TGA still gives the smallest error from an $m$-term approximation if we allow greedy sums to be of size $\lceil \lambda m\rceil$, and 2. not $1$-strong partially greedy but $1$-($\lambda$, partially greedy) for some $\lambda > 1$. Finally, we prove various equivalences for different greedy-type bases.