A possible method to consistently reach the necessary caster level to both gain and use the feat Craft Staff is to become a 5th level Red Wizard of Thay, allowing the character to become a circle leader and to regularly reach higher caster levels for 24 hours per day.
While your question initially wanted to limit the involvement of characters to only a single PC, I believe that such a proscription might leave the answer set empty. While this answer does require the existence of helpful NPCs (or otherwise very helpful PCs), this all appears to be legal for a typical E6 game.
You'll have to die
But, first you'll need to acquire at least five levels in a class that has Spellcraft as an in-class skill, and be sure to invest at least the required 8 ranks into Spellcraft. You'll also need to acquire the feat Tattoo Focus, and any three other metamagic or item creation feats.
For this reason I do recommend starting as a human wizard, for whom these feat requirements are easily achieved by even 3rd level, without the need for flaws.
Once your character has the needed feats and at least 8 ranks in Spellcraft, seek out the services of a 5th or 6th level artificer. An artificer of such level will be able to produce scrolls of Reincarnate. You'll need at least 4, possibly 5 scrolls of Reincarnate. Then, you'll need to be killed and reincarnated enough times to reduce your level to one. Along the way, your character will retain "any class abilities, feats, or skill ranks it formerly possessed." (PHB p.270)
Now, just start over
Progressing from a character of level 1, while possessing at least 8 ranks in Spellcraft, the feat Tattoo Focus, and three metamagic or item creation feats.
At each next level, be sure to select the Red Wizard of Thay as your character class, until you are a Wizard 1/Red Wizard of Thay 5. (You could also be a Cleric 1, or a Sorcerer 1, or really any character with a caster level of 1 at first level.)
Such a methodology of retraining might be incredibly wasteful and inelegant, but is supported by the rules.
At reaching Red Wizard of Thay 5, you'll have the ability to lead a circle. Circle benefits, among others, allow the circle leader to increase their caster levels for 24 hours at a time - acceptable for feat qualifications and for item creation.
You'll need to acquire the services of at least two additional spellcasters who have the feat Tattoo Focus and who also don't mind contributing a 3rd level spell for every day that you plan on using Craft Staff, so this answer (like WannabeWarlock's) still might require the use of an "organization" or "collective" of one kind or another.
Red Wizard doesn't really boost caster level, right?
Red Wizard's spell power class feature, as well as the wording of the circle magic effect, makes clear that the Red Wizard's caster level are increased for level-dependent variables, as well as caster level checks made to overcome spell resistance.
However, that seems to only be restating the general rule on what caster level increases mean (PHB p.171).
In the event that a class feature, domain granted power, or other
special ability provides an adjustment to your caster level, that
adjustment applies not only to effects based on caster level (such as
range, duration, and damage dealt) but also to your caster level
check to overcome your target’s spell resistance (see Spell Resistance, page 177) and to the caster level used in dispel checks (both
the dispel check and the DC of the check).
Indeed, a tight enough reading of such a rule would seem to greatly ameliorate almost all caster level boosting effects that do not come from items only. (That is, for example, a very "correct" reading of the above rules would seem to make the Ultimate Magus' spell power just the same as the Red Wizard's ... seemingly inapplicable for this purpose.)
While such a reading of what it means to boost caster level does tend to create a conversation that is fractious and ill-defined, it seems that, for this questioner and his or her DM, Ultimate Magus does work to enhance caster level.
So, I would put forward that such a DM ought be open to the caster level boosting effects of the Red Wizard being read the same way.
Finally, you can use the Silverymoon's College or a Red Wizard
Perhaps, incongruously, I was having some trouble imagining how the Lady's College would provide the services necessary for Craft Staff in an E6 world when none of the students or instructors (if this question is any indication) had themselves yet figured out how to Craft a Staff.
It might be the case that they, occasionally or otherwise, contract out the services of a rare high level Red Wizard (such as the one described in this answer) to be able to make available the higher level item creation feats normally unavailable in an E6 game.
Of course, we're really subverting the rules for E6
In retrieving and reading this document regarding E6, it would appear that attempts like this to be able to craft higher level items run against what the "designers" (if designer is even an applicable word here) of E6 wanted.
Q: Can you make high-level items as a low-level caster in E6?
The remaining portion of the answer indicates a "why" it should be that case, but since we're already subverting that, I offer this closing tidbit out of a sense of completeness instead of a remonstration of the question and my answer.