A well-known result in [Hsu-Smith-Waltman, Trans. Amer. Math. Soc. (1996)] states that in a competitive semiflow defined on
X
+
=
X
1
+
×
X
2
+
X^+ = X_1^+ \times X_2^+
, the product of two cones in respective Banach spaces, if
(
u
∗
,
0
)
(u^*,0)
and
(
0
,
v
∗
)
(0,v^*)
are the global attractors in
X
1
+
×
{
0
}
X_1^+ \times \{0\}
and
{
0
}
×
X
2
+
\{0\}\times X_2^+
respectively, then one of the following three outcomes is possible for the two competitors: either there is at least one coexistence steady state, or one of
(
u
∗
,
0
)
,
(
0
,
v
∗
)
(u^*,0), (0,v^*)
attracts all trajectories initiating in the order interval
I
=
[
0
,
u
∗
]
×
[
0
,
v
∗
]
I = [0,u^*] \times [0,v^*]
. However, it was demonstrated by an example that in some cases neither
(
u
∗
,
0
)
(u^*,0)
nor
(
0
,
v
∗
)
(0,v^*)
is globally asymptotically stable if we broaden our scope to all of
X
+
X^+
. In this paper, we give two sufficient conditions that guarantee, in the absence of coexistence steady states, the global asymptotic stability of one of
(
u
∗
,
0
)
(u^*,0)
or
(
0
,
v
∗
)
(0,v^*)
among all trajectories in
X
+
X^+
. Namely, one of
(
u
∗
,
0
)
(u^*,0)
or
(
0
,
v
∗
)
(0,v^*)
is (i) linearly unstable, or (ii) linearly neutrally stable but zero is a simple eigenvalue. Our results complement the counterexample mentioned in the above paper as well as applications that frequently arise in practice.