We study the global-in-time well-posedness and relaxation limit of the compressible Euler system with damping in
L
p
L^p
-type critical Besov spaces. In comparison with the results obtained by Crin-Barat and Danchin [Pure Appl. Anal. 4 (2022), pp. 85–125; Math. Ann. 386 (2023), pp. 2159–2206], the more general pressure law satisfying
P
′
(
ρ
¯
)
>
0
P’(\bar {\rho })>0
is allowed. To achieve it, a new composition estimate is established in the
L
2
L^2
-
L
p
L^p
hybrid Besov spaces with explicit dependence on the threshold between high frequencies and low frequencies.