  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 $\phi_f(v)$ the sum of
$f(e)$ over all edges $e$ incident to $v$. If $\phi_f(v) \neq \phi_f(u)$ 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 $\tau(G)$ to be the
maximum integer such that $G \cup tP_3$ is antimagic for all $t \leq \tau(G)$.
Chang, Chen, Li, and Pan showed that for all antimagic graphs $G$, $\tau(G)$ 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 $\tau(G)$ 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 $\tau(G)$ for these special families. In the present
paper, we generalize their results by proving an upper bound on $\tau(G)$ 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 $2C_3$.
