  We show that the moduli space of $U\oplus \langle -2k \rangle$-polarized K3
surfaces is unirational for $k \le 50$ and $k \notin \{11,35,42,48\}$, and for
other several values of $k$ up to $k=97$. Our proof is based on a systematic
study of the projective models of elliptic K3 surfaces in $\mathbb{P}^n$ for
$3\le n \le 5$ containing either the union of two rational curves or the union
of a rational and an elliptic curve intersecting at one point.
