The directed polymer model describes random paths under the influence of a space-time random environment. We consider the high-temperature, weak disorder regime in spatial dimension $d\geq 3$, where it is known that the paths obey a central limit theorem. Specifically, we focus on the sub-regime where the associated martingale $(W_n)_n$ is not $L^2$-bounded and analyze the spatial correlations of the field $(W_n^x)_x$, where the superscript $x$ indicates the starting location of the polymer. In the case of a bounded environment, we show that a suitably re-centered spatial average over a set of diameter $n^{1/2}$ converges to zero at rate $n^{a+o(1)}$, with an explicit exponent $a$ that is different from the corresponding exponent in the $L^2$-bounded case.