Increase the first argument of the expand = argument of scale_x_discrete:
ggplot(glomor, aes(x=Time.Period, y=Score, group=Variable, shape=Variable,
color=Variable, width=0.5)) +
geom_point() +
scale_x_discrete(limits = c("Jan 1997 - Aug 2003",
"Sep 2003 - Jun 2005","Jul 2005 - Dec 2009"), expand=c(0.2, 0)) +
geom_line(aes(linetype=Variable), size=1.3) +
geom_point(size=4.4) +
scale_shape_discrete(name="", labels=c("Significant Z-Score", "Moran's I Z-Score")) +
scale_linetype_manual(values=c(1, 3), name="",
labels=c("Significant Z-Score", "Moran's I Z-Score")) +
scale_color_manual(values=c("black", "grey40"), name="",
labels=c("Significant Z-Score", "Moran's I Z-Score")) +
scale_fill_discrete(name="", labels=c("Significant Z-Score", "Moran's I Z-Score")) +
theme_classic()+
ylim(0,10) +
xlab("Time Periods") +
ylab("Moran's I Z-Score") +
theme(axis.text.x=element_text(size=11)) +
theme(axis.title.x=element_text(size=14, face="bold")) +
theme(axis.title.y=element_text(size=14, face="bold")) +
theme(legend.position=c(.80, .90)) +
theme(legend.text=element_text(size=12)) +
theme(legend.background = element_rect(fill="white", linetype="dashed")) +
theme(legend.key = element_blank())