The level of residual chlorine in tap water is below what you'd likely smell (and the smell usually attributed to chlorine is actually the reaction products of chlorine and organic materials, of which there should be little to none in tap water, that's why swimming pools smell, chloramines are pungent). I've sniffed actual unadulterated chlorine so you don't have to, but it doesn't smell like bleach, but it does smell like ouch*. The taste in tap water is usually attributable to the mineral content (in part from the source, leaching from pipes, and flocculents etc. used in processing). Cooling will simply reduce the taste. We have a filter in our fridge, I mostly don't bother (the pipework also goes mouldy quite quickly, I'd rather have mould-free direct from the tap thanks.
*one of my many ian-does-stupid-for-money student jobs was bleaching flour**, which is done with either chlorine gas (made in a special machine, that I was the completely underqualified captain of) or peroxide, which was too safe for the place I worked, desperate students being willing to do anything.
**flour mills are awesomely dangerous, flour dust is extremely explosive, there are dangerous chemicals everywhere, and the rats are the size of cows, and the noise they make when they go through the rollers will haunt your dreams.
