Here is the same exercise you saw earlier only for the 6th 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 5th 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 6th string only with different fingers.
Once you have learned the names of the notes and the correct fingerings for the 5th string and the 6th string, practice one and then the other.
If you decide not to learn the other strings it is important to know the names of the notes for 6th and 5th strings. Why? These 2 strings are the basis for moveable chord and scale patterns so that you can play up and down the neck.
For example, if a song is in the key of A major then the common chords played are A - D - E or E7. Thus, the song is based on these fretboard notes;
If you change key to say Ab the patterns are all the same only everything is moved down 1 fret. This allows you to play in any key once you know the names of the notes on the 6h and 5th string along with the chord/scale patterns.
The scale patterns will allow you to solo with the chords without hitting any notes that might sound like "Duck Calls".