Maybe it's something to do with air/chlorine in the water as it comes straight out of the tap and then this gas dissipates after 30 mins.
This is a definite possibility.
Anything that remains in suspension in a liquid such as yeast or trub can affect the SG because it's effectively changing the density of the liquid. The effect may be slight or noticeable depending on how much is in suspension. Gas has a similar effect, whether it is CO2, oxygen, or whatever because it affects density. Also, bubbles that cling to the hydrometer can cause it to float better.
Therefore gas absorbed in water can make it less dense and cause the hydrometer to sink lower, whereas gas escaping from the water can cling to the hydrometer and cause it to float more.
Your hydrometer is likely only accurate at 20°C in non-fizzy beer that has most yeast and trub dropped out of suspension.
Taking readings of wort prior to fermentation must be done cooled to 20°C, but even then will be affected by particles of grain in the wort. You are rarely going to get a precise reading, but if you accept the margin of error and remove as many variables as you can then a hydrometer is probably good enough for home brew purposes.
Repeated dipping or spinning the hydrometer in your liquid helps remove bubbles and can make a difference of about 2 points on your SG reading, I have found.