How about this:
First find a cover of $E$ which has sum of lengths less than 1/2,
The find another cover of $E$ which has sum of lengths less than 1/4,
The union of all the covers will then be a cover with sum of lengths less than 1, and we just need to show that each point of $E$ is in an infinite number of the covering intervals (it might happen that some of the intervals chosen at each step are the same as intervals chosen at a previous step).
To prove this last point, take a point $x$ and suppose it is contained in only finitely many (say $n$) of the intervals. Let the length of the smallest of these be $y$. Then there is an interval in a "later covering" (of total length less than $y$) containing $x$, and it must be of length smaller than any of the $n$ intervals. This gives a contradiction.