Let
(
R
,
m
)
(R,\mathfrak {m})
be a Noetherian local ring of dimension
d
≥
2
d\geq 2
. We prove that if
e
(
R
^
r
e
d
)
>
1
e(\widehat {R}_{red})>1
, then the classical Lech’s inequality can be improved uniformly for all
m
\mathfrak {m}
-primary ideals, that is, there exists
ε
>
0
\varepsilon >0
such that
e
(
I
)
≤
d
!
(
e
(
R
)
−
ε
)
ℓ
(
R
/
I
)
e(I)\leq d!(e(R)-\varepsilon )\ell (R/I)
for all
m
\mathfrak {m}
-primary ideals
I
⊆
R
I\subseteq R
. This answers a question raised by Huneke, Ma, Quy, and Smirnov [Adv. Math. 372 (2020), pp. 107296, 33]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of
I
I
.