Let $X_1, \dots, X_n$ be iid zero mean and unit variance random variables with some $s > 2$ moments. Then by KMT results, essentially $$ \left|\sum_{t=1}^{n} X_t - W(n) \right| = O(n^{1/s}) \quad a.s. $$
Now suppose I look at the partial sum of a chunk of the random variables, where the size of the chunk increases with $n$. Let $c_n$ be the size of the chunk and define the partial sum as $$ S_{c_n} = \sum_{t=n - c_n + 1}^{n}X_t\,. $$
Then intuitively, since I'm summing only $c_n$ random variables I should be able to say that: $$ \left|S_{c_n} - W(c_n) \right| = O(c_n^{1/s})\quad a.s. $$ but I'm not sure how to present a clean argument as the membership fo the partial sums keeps changing with $n$. Certainly, by writing the partial sum a difference between two chunks both starting from 1, I can get a $O(n^{1/s})$ bound, but it seems reasonable that a $O(c_n^{1/s})$ should be viable.
As noted by the OP, "The membership of the partial sums keeps changing with $n$".
In fact, this implies that the $O(n^{1/s})$ bound is essentially optimal (up to logarithmic corrections.) Indeed, suppose $X_j$ satisfy $E|X_j|^s<\infty$, but $E|X_j|^{r}=\infty$ for all $r>s$, see [1]. Suppose That $|S_{c_n}|<Cn^{1/r}$ for some $r>s$ and all $n$. Then it follows that for all $n$, we have $|Y_n|<2Cn^{1/r}$, where $Y_n$ is either $X_{n+1}$ or $X_{n+1}-X_{n-c_n}$. But $$\sum_n P(|Y_n|>Cn^{1/r}) = \sum_n P(|Y_n|^r>C^r n) >c_1 E|X_1|^r =\infty\,,$$ so by the Kochen-Stone version [2] of the Borel-cantelli lemma, $$P(|Y_n|>Cn^{1/r} \quad \text{infinitely often})=1 \,.$$
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[1] For example, $X_j$ could have have density $$f(x)=c(|x/a|+1)^{-s-1}[\log(|x/a|+2)]^{-2}\,,$$ where $a$ and $c$ are chosen so that $\int_{-\infty}^{\infty}f \, dx=1$ and $\int_{-\infty}^{\infty}x^2 f(x) \, dx=1$.
[2] https://proofwiki.org/wiki/Kochen-Stone_Borel-Cantelli