Don't worry about getting your pick to halfway between strings before you change direction, just make sure you've hit the string, then start heading back.
On the pick - make sure you hold it at an angle to the strings. Mine is probably about 45 degrees, but experiement on what feels right for you - you can pick faster with it angled. Also, make sure you have a minimal amount out past your fingers so you don't get caught up in the strings.
String skip. If you're doing spider exercise 1234,1234,1234 type exercises, do the same thing but miss a string, or maybe miss two, or three or whatever. It'll help you locate the strings when you're not just moving between adjacent ones. Arpeggiating chords might help with this one too.
Finally - don't look at your right hand while picking. Well, maybe spend some time looking, but some time looking away as well so you can feel where the strings are instead of looking.
Apologies if any of that is obvious, as you've been playing 10 years. If you're still having trouble, chances are you might need to go back a step to move forward.
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