Jeff, my brother!
I applaud and support your desire to work on the classic pass.
Break management is an important aspect for all of your card handling, especially for the classic pass. The moment a card is replaced to the deck your audience will be watching extremely close! Not because they know what to look for, rather, because if any secret action is to happen, they expect it to happen right now. This moment, after the card is returned, but before they're convinced you're not exercising control over it, is what Darwin Ortiz refers to as the "Critical Interval." We shouldn't be doing any sleights during this time. More on this in Ortiz's book Designing Miracles.
For the classic pass, the fourth finger break acts as a time displacing tool. It allows us to wait for better moments, for the off beat. After all, we can obtain a fourth finger break much easier than we can execute a pass under an intense, watchful eye, and by using a break we don't have to execute our pass under such conditions. That being said, we will lose the battle before we've pulled out the big sleights if we're caught holding a break.
Our audience only needs to see the two halves still separated, the break, in order for the entire sleight to be seen. They will not care for how we controlled their card if they know that we never truly lost control of their card.
Now we arrive at your question. How do you eliminate the slight brief at the front of the deck?
The simple answer is to use a jog instead of a fourth finger break. An injog will mostly be interchangeable with a fourth finger break and, in the case of the classic pass, will allow for a more casual handling of the cards. Then, once you're ready to execute your pass, your hands converge and convert the injog to the required break.
Should you still be inclined (and you should be inclined) to better your break, start by placing the cards in mechanic's grip. Having the first finger at the front of the deck will cover quite a bit of that line you have forming at the front of the deck. Whatever remains visible will more than likely be the fault of your thumb, second and fourth finger. The break held by the fourth finger should be by the flesh of the pad of your finger, not the tip. Your thumb and second finger work together by applying pressure to their respective sides. Chances are, if you are already holding a rather minute break, your second finger and thumb aren't squeezing enough to keep the two halves together. By applying proper pressure to the sides of the deck I'm certain you'll notice some improvements.
Of course, you can also square the ends a bit more with your right hand if it remains visible. Simply put: Mechanics grip, flesh break, proper pressure at the sides.
Let me know if any of this doesn't read clearly and I'll be more than happy to elaborate.
Respectfully yours,
Cameron J. Braxton