  In this article we study local constancy of the mod $p$ reduction of certain
$2$-dimensional crystalline representations of
$\mathrm{Gal}\left(\bar{\mathbb{Q}}_p/\mathbb{Q}_p\right)$ using the mod $p$
local Langlands correspondence. We prove local constancy in the weight space by
giving an explicit lower bound on the local constancy radius centered around
weights going up to $(p-1)^{2} +3$ and the slope fixed in $(0, \ p-1)$
satisfying certain constraints. We establish the lower bound by determining
explicitly the mod $p$ reductions at nearby weights and applying a local
constancy result of Berger.
