  This paper revisits the problem of multi-server Private Information Retrieval
with Private Side Information (PIR-PSI). In this problem, $N$ non-colluding
servers store identical copies of $K$ messages, each comprising $L$ symbols
from $\mathbb{F}_q$, and a user, who knows $M$ of these messages, wants to
retrieve one of the remaining $K-M$ messages. The user's goal is to retrieve
the desired message by downloading the minimum amount of information from the
servers while revealing no information about the identities of the desired
message and side information messages to any server. The capacity of PIR-PSI,
defined as the maximum achievable download rate, was previously characterized
for all $N$, $K$, and $M$ when $L$ and $q$ are sufficiently large --
specifically, growing exponentially with $K$, to ensure the divisibility of
each message into $N^K$ sub-packets and to guarantee the existence of an MDS
code with its length and dimension being exponential in $K$. In this work, we
propose a new capacity-achieving PIR-PSI scheme that is applicable to all $N$,
$K$, $M$, $L$, and $q$ where $N\geq M+1$ and $N-1\mid L$. The proposed scheme
operates with a sub-packetization level of $N-1$, independent of $K$, and works
over any finite field without requiring an MDS code.
