It is given that fights in general and each individual round, by which fights are scored, always start from the outside. It's not a coincidence that there are literally a dozen variations of the jab to serve different purposes regardless of your style, making it the most diverse and versatile punch in the toolbox. It's your longest punch, most accurate, and fastest punch when you're in your stance. It's the most suitable punch to use for finding your range, keeping your opponent occupied (while you're dictating as the busier man), managing distance, creating openings, etc. It requires the least amount of commitment which means that it's generally the safest punch to throw. Saying that, it happens to be the punch most countered largely because it's thrown most frequently (exponentially more), disproportionately to the rest. Though, when it is countered, it's often down to being thrown either lazily or predictably.
Besides the jab, you can utilize a number of feints to set up leads; the jab feint specifically, a foot or shoulder feint (either shoulder), and so on. You can also utilize fakes. Fakes involve less commitment than a feint would but don't "sell" as well. Quick level changes work too (they can also serve as feints), as can dips (get your head off the line before firing your lead), and footwork to advantageously position yourself to more safely throw leads that aren't set up by the jab or by feints/fakes. With the latter you'd optimally want a clean angle on your opponent to minimize or even eliminate any exposure by reducing your attack surface prior to shooting your leads.
If you wanted to cheat a bit on the speed of your rear hand leads then you can stay in your stance but square your upper body up to throw them faster (power will be sacrificed here for speed). You can also play with the weight distribution by staying heavy on the front foot to increase their speed (your weight is pre-loaded to compensate for the extra time it would normally take to transfer it forward).