Let $C$ be a reduced (not irreducible) projective curve of degree $d$ such that $C$ contains at most double points. By a result due to Kleiman and Altman, we know that there exists a smooth surface containing $C$. Can we compute $n$ (explicitly) such that for any $m \ge n$, there exists a smooth degree $m$ surface containing $C$? What happens if we assume that $C$ is smooth?
I was trying to solve this question in the following way: If I understand correctly, Theorem $7$ in "Bertini theorems for hypersurface sections containing a subscheme" by Kleiman and Altman, states that $C$ can be embedded in a hypersurface in $\mathbb{P}^3$ (because if the "embedded dimension" $e(C)$ is equal to $1$ (resp. $2$) then the dimension of $X_e$ is equal to $1$ (resp. $0$) by definition) and also gives a rough estimate for the value of $n$ stated in the question. But there should be an error in my understanding because if I take $C$ to be of the form $\cup C_i$ where $C_i$ are $-2$ exceptional curves and $C_i.C_j \le 1$, then it cannot be embedded into a smooth hypersurface in $\mathbb{P}^3$ of degree not equal to $4$ (due to contradiction with the adjunction formula). Could someone point out precisely, where I am getting it wrong?
Note $X_e$ consists of the set of points on $C$ with embedded dimension $e$.