To quickly and effectively make your way around lesser-traveled areas of the map, it all comes down to situational awareness and quick thinking.
If you play a stealthier scout and need to part from your team to flank the enemy, you'll need to be able to scan your surroundings and follow paths that lead away from the main path. You can use BONK! Atomic Punch to give yourself a third-person view, which will allow you to see past corners by angling yourself without actually exposing yourself - this allows for staying hidden while traversing the other parts of the map. Alternatively, take advantage that you are the fastest class and find a hiding spot (or get ready to jump over the enemy to confuse them) so that you may slip away or attempt to make a kill.
Another thing to note is that you should never use the Baby Face Blaster. This is because it slashes the scout's default speed and conditions you to avoid using your air jump and keeps you on the ground, and in many cases your air mobility will be the difference between life and death.
When exploring the map, keep an eye for any buildings you can travel through/around and any other area that looks like it would be ideal for a sniper to hide or for a spy to uncloak - in many cases, the enemy team will not be checking this area unless they are poorly coordinated (walking about everywhere), under pressure from a target in said area, or choosing to roll-out through an alternative route.
To answer your question of strategy: When your team sticks together, they will be the primary focus of the enemy, allowing you to slip past. Make use of double jumps for reaching higher places and avoiding the other team.
As I mentioned before, it is a sin for a mobile scout to use the Baby Face Blaster. I mentioned that it conditioned you to stay on the ground and avoid the use of your double jump, and how it cuts your speed. From the get-go, equipping the Baby Face Blaster will severely slow down your roll-out, as you will not be crossing the map as fast as you potentially could.
If your playstyle revolves heavily around picking off lone enemies who don't know you are there, I would recommend using the Backscatter. At maximum range for its minicrit effect, it can deal 80 damage if every pellet hits (8 damage per pellet, 10 pellets per shot total). At point blank, it deals a significantly higher damage, a total of 140 if every pellet hits (14 damage per pellet, 10 pellets total) - this can instantly kill an enemy scout, engineer, sniper, or spy if they are not using equipment that would put their health over 140.
As always, the stock Scattergun is always a great option too.
For your style, I would recommend three pistols: stock, the Winger, and the Pretty Boy's Pocket Pistol. The Winger can quickly be pulled out to deal more damage than the stock pistol per shot, but cannot be relied on due to its clip size being less than half of that of the stock pistol. The extra jump height can come in handy for climbing the map as scout, but will not truly help in a close-quarters battle. The Pretty Boy's Pocket Pistol grants buffs such as never taking fall damage and 3 health on hit while the weapon is active, at the cost of 20% damage vulnerability. The health on hit can help you heal up when your enemies are distracted, although you need to be a little more careful when you pull it out due to the 20% damage vulnerability when it's active.
Your only options for the bat should be stock or the Sandman, as you will likely be travelling alone and need to maximize your damage output. The Sandman is good for stunning an enemy that's aware of you to take advantage of their helpless phase, but when used on an unsuspecting enemy it could be the difference between a high-damage minicrit shot to the back and a normal-damage shot to their face. The Sandman also lowers your maximum health, meaning it will make you less likely to survive getting shot if you attract the attention of the enemy team.