Add to default text colour palette

Unfortunately this is a limitation of the default WP text color picker. You may notice that the same happens also on pages/posts and not just for widgets.

I’d recommend to use the TinyMCE Color Picker plugin, which saves the custom colors in a persistent way.

I tried the TinyMCE Color picker, and it is ALMOST exactly what I need, however, you can’t manually enter hex codes. Any ideas on how to do this?

The current version of TinyMCE Color picker allows to add colors by entering the hex code. You just have to click on the “Add a color” button.
For further assistance I’d suggest to post to “TinyMCE Color picker” support forum.