What techniques do you use to help develop the plot alongside gameplay when you finally have an idea for a story down?

There are many ways to build games.

As a direct answer to your question:  Virtually none.  Basic gameplay mechanics should be fleshed out with only the most vague ideas of a story. Mechanics should be identified which are fun and compelling in their own right, not forced into a story.

Longer answer:

In my experience in the professional world, the story tends to be far down the list and relatively late in the process.  Often the people writing the game are told a general theme and told the rest will be fitted to the mechanics.

There are mechanics and gameplay experiments long before stories are ever considered.  Developers experiment with the mechanics, try a wide range of mechanics, try to find ways to make the mechanics work together or break when used in combination. If there is a hint of a story at this point it is only the most vague framework to explain how the mechanics interconnect, more of providing a genre rather than plot points.

I've worked on and worked with others on many projects where the studio builds mechanics to be played with based only on the most vague specifications.  We have an early project like that right now at my job where a small group of people have been assigned to implement a range of mechanics for games that don't even have a genre.  Questions like "Is this going to be more of a tactical game or more of a strategy game?" are answered with "here are some more mechanics that look good on paper mockups, depending on how they feel at computer speed we might be able to answer that."  

As a large suite of mechanics is developed the beginnings of a story can be built that can leverage the mechanics. As the mechanics are being pruned down to the essence of the game, the story pieces can start to form built around the mechanics. A full plot should wait until the mechanics are realized and gameplay is fun.  A fun game system can be used to tell many varied stories, but good stories tend to be specific to the game systems they are in.  And many amazing games have no story at all.

In my observations of the hobby world, the many games based around stories will write the story and the plot, they'll have all kinds of numbers for their supposed tuning long before any code is written. 

I've seen design notebooks filled with specific game items, specific armors, specific weapons, all with a long list of (meaningless) values. Often there is a teen-focused adventure story full of plot twists and bad writing where the teen overcomes the school bully and vanquishes the world. There are drawings of the characters, their faces, descriptions about how charismatic or crude each party member should be.  There are intricate sketches of weapons and armor, and full descriptions of how Epic Armor of Awesomeness has 1337 defense plus 40% defense against undead monsters and school administrators, but there is no definition of the math those numbers plug into.

These projects tend to fall down long before code is ever written or actual game art is ever created. A few will have some experimental code written around them, or some white-box levels written in Unity or Unreal. But when they realize just how much work is involved, that they've created a project that would require thousands of work-years to implement, it will be abandoned by the wayside.

I fully agree about hobby games that tackle complex or new gameplay mechanics. These should go on the back burner until the hobbyist has a lot of experience.

Twine/Inklewriter games, visual novels, and RPG Maker games, however, are within the reach of even first-time hobbyists, and they're a good way to hone interactive writing skills as well as general game design skills.

So when you finally have an idea for a story down, you might consider choosing an existing gameplay genre (such as a visual novel) that fits your story and your current level of experience.