Let $X$ be a smooth plane projective quintic curve (over $\mathbb C$). Then we know that it has gonality $4$. Assume that it has genus $g(X)=6$. Then my question is the following:

Is it necessarily true that for every line bundle $A$ on $X$ with $h^0(A) \geq 2$ one has $\text{deg}(A)\geq h^0(A)+2$?

Gonality $4$ means minimum degree of line bundles with atleast $2$ sections is $4$. On the other hand we have for any line bundle with atleast $2$ sections and with $h^1(A) \geq 2$, $\text{deg}(A) \geq 2h^0(A)-2$. But then it's not quite clear to me that how these two facts on gonality and clifford index (and may be Riemann-Roch) give us an affirmative answer to the question. Maybe I'm missing something obvious. Does there exist a more direct proof in the literature?