Let $G$ be a graph with $m$ edges and let $f$ be a bijection from $E(G)$ to $\{1,2, \dots, m\}$. For any vertex $v$, denote by $φ_f(v)$ the sum of $f(e)$ over all edges $e$ incident to $v$. If $φ_f(v) \neq φ_f(u)$ holds for any two distinct vertices $u$ and $v$, then $f$ is called an {\it antimagic labeling} of $G$. We call $G$ {\it antimagic} if such a labeling exists. Hartsfield and Ringel in 1991 conjectured that all connected graphs except $P_2$ are antimagic. Denote the disjoint union of graphs $G$ and $H$ by $G \cup H$, and the disjoint union of $t$ copies of $G$ by $tG$. For an antimagic graph $G$ (connected or disconnected), we define the parameter $τ(G)$ to be the maximum integer such that $G \cup tP_3$ is antimagic for all $t \leq τ(G)$. Chang, Chen, Li, and Pan showed that for all antimagic graphs $G$, $τ(G)$ is finite [Graphs and Combinatorics 37 (2021), 1065--1182]. Further, Shang, Lin, Liaw [Util. Math. 97 (2015), 373--385] and Li [Master Thesis, National Chung Hsing University, Taiwan, 2019] found the exact value of $τ(G)$ for special families of graphs: star forests and balanced double stars respectively. They did this by finding explicit antimagic labelings of $G\cup tP_3$ and proving a tight upper bound on $τ(G)$ for these special families. In the present paper, we generalize their results by proving an upper bound on $τ(G)$ for all graphs. For star forests and balanced double stars, this general bound is equivalent to the bounds given in \cite{star forest} and \cite{double star} and tight. In addition, we prove that the general bound is also tight for every other graph we have studied, including an infinite family of jellyfish graphs, cycles $C_n$ where $3 \leq n \leq 9$, and the double triangle $2C_3$.