Understand it is a Rule of Thumb
So, 6 encounters is a rule of thumb, not a hard and fast "you must make level X after 6 encounters." Also, keep in mind that when it comes to balance and leveling some of the advice in the DMG is hard to follow to the letter. I for instance, have never been able to get the "6 to 8 encounters a day" -- so instead I only use "hard" or "deadly" encounters, which I find my players seem to manage too easily most of the time. They aren't minmaxed, they just play to their strengths in the actual fight.
Likewise for your group, changing the way XP is totaled, or ignoring XP for level purposees might make sense. Let talk about this though.
Not all Encounters are Combat
The books don't talk about the proper amount of XP to give to social encounters, trap encounters, puzzle encounter, etc. But the books and game designers do state that XP should be awarded for them. The reason there aren't charts for them is because they are hard to judge without actually being there. So, if you keep battle XP calculated the same, but start giving XP for these other types of encounters, you might reach the suggestions in the DMG.
For Battle Encounters
The battles your players face will have various numbers of foes. Fewer foes mean larger monsters. This means lower or no multiplier, so the split is better for the players. So, throw in a solo "boss" monster here and there. Or a boss monster with "squishy" minions.
Another option is to look at the published adventures, even if you don't plan to run them. For example Lost Mine of Phandelver, it tell you how many monsters will be in each area. It is calculated that if you run the option first quest the average party should gain one level, before you begin the main story arcs.
How they do manage this? They do a few encounters with a lot of goblins, but by the end you are basically in a mini-boss fight. They've done the math for each encounter ahead of time, and know that running that part will grant X XP for this encounter with a part of Y players.
"Would it be ok if I started adding the multiplier to the actual experience gained?"
Yes. The beauty of the game is that the rules are a guide post. If you do start adding the multiplier, it won't break anything -- but it will make your characters level faster, which will change the feel of the game as character progression would be sped up, but some people like that. If you do, just keep in mind to balance encounters so that they are on the harder side.
Another Option: Milestones
I personally don't use XP. I use milestones. It is more subjective, but it feels more natural to me. With XP, a big boss fight might put you a few points away from the next level, and end up leveling on a random goblin -- might feel anti-climatic. Where with milestones, you kill the dragon -- welcome to the next level.
This also allow the DM to control the pace of the leveling. If you need them to be a certain level for a certain encounter, you can get them there before they reach it.
Question in the title: How to get from 1 to 2 in 6 encounters.
Assume party size of 4 players:
Each player has to get 300 XP. 300 X 4 = 1200 XP.
(using Kobld Fight Club for ease) I generate 6 random encounters... Say 1 easy, 2 medium, 2 hard and 1 deadly. I get:
- 1 Ape (easy) 100XP
- 1 spy (medium) 200XP
- 1 Cultist, 3 Lizard, 2 Mule (medium) 105XP
- 1 Blue Faerie Dragon (hard) 450XP
- 1 Merrow (hard) 450XP
- 1 Carrion Crawler (deadly) 450XP
Total 1,755 XP. Well over the needed 1200 XP. In fact enough XP that a party of 5 players could all make level 2.