What ulrich writes is correct. I will delete this answer if ulrich posts his answer. For me, the fastest way to think about this is via smoothness and irreducibility of the stack of locally free sheaves of specified rank and degree on a curve $X$.

For a fixed smooth, projective curve $X$ over a field $k$, there is an algebraic stack $\text{Bun}_{r,d}(X)$ parameterizing locally free $\mathcal{O}_X$-modules of rank $r$ and total degree $d$. This stack is smooth, since the obstruction group $\text{Ext}^2_{\mathcal{O}_X}(E,E)$ is zero. The stack is not quasi-compact (the slopes in the Harder-Narasimhan filtration of $E$ can be arbitrary integers subject to the constraint coming from $r$ and $d$). However, every quasi-compact open substack $V$ is integral in the sense that there is a representable, fppf (even smooth) $1$-morphism $p:V_0\to V$ from a (geometrically) integral $k$-scheme $V_0$. This follows, for instance, by writing every bounded family of locally free sheaves $E$ as quotients of some fixed locally free sheaf $\mathcal{O}(-N\cdot A)^{\oplus (r+1)}$ where $A$ is a fixed divisor class on $X$ of positive degree, and $N\gg 0$ is a sufficiently positive integer. In that case, $V_0$ is an open subset of an affine space bundle over the Jacobian of $X$.

For $X$ an elliptic curve, there is an open substack $U\subset \text{Bun}_{2,0}(X)$ parameterizing locally free sheaves of the form $E=L\oplus M$ where both $L$ and $M$ are invertible sheaves of degree $0$ such that $L\not\cong M$, e.g., $L\oplus \mathcal{O}_X$ where $L\not \cong \mathcal{O}_X$. The key to openness of this substack is the fact that every infinitesimal deformation of $L\oplus M$ is again of the form $L'\oplus M'$, since $$\text{Ext}^1_{\mathcal{O}_X}(L,L)\oplus \text{Ext}^1_{\mathcal{O}_X}(M,M)\to \text{Ext}^1_{\mathcal{O}_X}(L\oplus M, L\oplus M)$$ is surjective. This uses the fact that $\text{Ext}^1_{\mathcal{O}_X}(L,M)$ and $\text{Ext}^1_{\mathcal{O}_X}(M,L)$ are zero on a genus $1$ curve for invertible $\mathcal{O}_X$-modules $L$, $M$ of degree $0$ such that $L\not\cong M$.

Because of integrality of $V_0$, the open subset $U$ is "dense" in $\text{Bun}_{2,0}(X)$: the inverse image of $U$ in $V_0$ is a nonempty open in the integral scheme $V_0$. Since the nontrivial extension of $\mathcal{O}_X$ by $\mathcal{O}_X$ gives a geometric point of the stack, for any geometric point of $V_0$ mapping to that point, there is a smooth affine curve in $V_0$ that contains that point and whose intersection with the inverse image of $U$ is dense.

**Edit addressing higher genus.** For a smooth, projective $k$-curve $Y$ of genus $g\geq 1$ such that there exists a flat $k$-morphism $f:Y\to X$, the pullback by $f$ of the genus $1$ example gives an example on $Y$.

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