Here's a weak variant that should really be a comment: For any twist $E^d: dy^2 = f(x)$ of the elliptic curve $y^2 = f(x)$, for either $E$ or $E^d$, there exist infinitely many primes such that your required condition is met.
Note that if $d$ is not a square mod $p$, that is, $E^d$ is a non trivial twist mod $p$, then $$E^d(\mathbb F_p)+E(\mathbb F_p) = 2(p+1).$$ This is because for every value of $x \in \mathbb F_p$, $f(x)$ is either a square in which case this contributes two points to $E(\mathbb F_p)$ or it is not a square, in which case it contributes two points to $E^d(\mathbb F_p)$. This shows that one of the two sets $E(\mathbb F_p)$ and $E^d(\mathbb F_p)$ has to be $\geq p+1$ and the other one has to be $\leq p+1$. Since there are infinitely many primes for which $d$ is not a square mod $p$, this proves the claim.