The OpenWeather's API this site uses doesn't keep the actual minimum temperature data of the day. Instead it changes the value to the minimum between the current hour and the end of the day.
This site only collects the minimum and maximum temperature once per day, so the later in the day you start using this site, the worse the effect explained above will be.
To circumvent this, you can simply show the next day's minimum. It shouldn't be a problem, right?