subbundle with $h^0(C, A)= 2$. Then $h^1(C, A) = 2- \deg(A) + g-1 \geq 2$, if and only if $g-1 \geq \deg(A)$, but this last inequality is satisfied by the semistability of $E$. Since $4 \leq h^0(E) \leq h^0(A) + h^0(A')  $, we get $h^0(A') \geq 2$.\
If $h^1(C, A')\geq 2$, then both $A$ and $A'$ contribute to the Clifford index. It follows that $$h^0(C, E) \leq h^0(C, A) + h^0(C, A') \leq \frac{\deg(A)-\mathrm{Cliff}(C) +2}{2} + \frac{\deg(A')-\mathrm{Cliff}(C) +2}{2}$$ $$=\mu(E) +2-\mathrm{Cliff}(C),$$ that is, inequality holds in that case.\
Suppose $h^1(C, A') \leq 1$. Applying the definition of Clifford index to the bundle $A$, we obtain $
\deg(A) \geq \mathrm{Cliff}(C) +2,
$ hence $h^1(C, A) = 2- \deg(A) +g-1 \leq g -\mathrm{Cliff}(C)-1 $. On the other hand, by means of the long exact sequence in cohomology, we have $$\begin{aligned}
h^0(C, E) &=& h^1(C, E) +d - 2(g-1) \\
& \leq& h^1(C, A) + h^1(C, A') + d -2(g-1) \\
& \leq & d - \mathrm{Cliff}(C) -g +2  \\
& \leq& \frac{d}{2} - \mathrm{Cliff}(C) + 2\end{aligned}$$ where the last inequality follows by the hypothesis on $d$.

From now on we shall assume that $E\in \cU_C(2, d)$ is globally generated and carries no sub-pencil. 