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 $3.8sigma$) 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.
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