Here is the same exercise you saw earlier only for the 3rd string.
Keep your thumb flat and positioned about the middle of the back of the neck. This will give your fingers a better reach and keep them from laying flat on the fretboard.
Try not to lift your fingers more than a 1/4" from the fretboard. This will increase your speed since the higher you lift off the longer it will take to move back down. This will also make it easier for you to get a feel for the neck so you don't have to always look at the fretboard.
Don't worry about speed. It is harder to play slow and in time than it is to play fast.
The 3rd String Exercise
Use the L.H. (Left Hand) fingerings shown below.
Play each note starting with open the string and move up the fretboard.
Say each note name as you play it.
Once you reach the 12th fret go back down in reverse order using the same fingerings & saying the note names.
Notice that you are using the same 3 hand positions as before with the other strings only with different fingers.
Once again when you have learned the names of the notes and the correct fingerings for the 3rd string add the 6th - 5th & 4th strings. Start with the 6th string then the 5th string, 4th and finally the 3rd.
Make sure that you always go up and then back down the fretboard.