  We present the results from our search for HI 21-cm absorption in a sample of
16 strong FeII systems ($W_{\rm r}$(MgII $\lambda2796$) $\ge1.0$ \AA\ and
$W_{\rm r}$(FeII $\lambda2600$) or $W_{\rm FeII}$ $\ge1$ \AA) at $0.5<z<1.5$
using the Giant Metrewave Radio Telescope and the Green Bank Telescope. We
report six new HI 21-cm absorption detections from our sample, which have
increased the known number of detections in strong MgII systems at this
redshift range by $\sim50$%. Combining our measurements with those in the
literature, we find that the detection rate of HI 21-cm absorption increases
with $W_{\rm FeII}$, being four times higher in systems with $W_{\rm FeII}$
$\ge1$ \AA\ compared to systems with $W_{\rm FeII}$ $<1$ \AA. The $N$(HI)
associated with the HI 21-cm absorbers would be $\ge 2 \times 10^{20}$
cm$^{-2}$, assuming a spin temperature of $\sim500$ K (based on HI 21-cm
absorption measurements of damped Lyman-$\alpha$ systems at this redshift
range) and unit covering factor. We find that HI 21-cm absorption arises on an
average in systems with stronger metal absorption. We also find that quasars
with HI 21-cm absorption detected towards them have systematically higher
$E(B-V)$ values than those which do not. Further, by comparing the velocity
widths of HI 21-cm absorption lines detected in absorption- and galaxy-selected
samples, we find that they show an increasing trend (significant at
$3.8\sigma$) with redshift at $z<3.5$, which could imply that the absorption
originates from more massive galaxy haloes at high-$z$. Increasing the number
of HI 21-cm absorption detections at these redshifts is important to confirm
various trends noted here with higher statistical significance.
