Option A: Use Monosyllabic Words instead of Numbers.
There are 10 digits in the English Language; so, sector 4-1-2 can give you a base-10 grid for 1000 sectors, averaging about 800,000 stars per sector across the Orion Arm.
Letters will get you a bit farther. Sector f-c-x can give you a base-26 grid with 17,576 sectors averaging 45,516 stars per sector grid... still pretty darn big.
But there is one thing that the English language has that is just as easy to say as a letter or number that we have a LOT more of: Monosyllabic words. In fact, there are about 12,000 words in the English Language that only have 1 syllable. This could be truncated down to the 10,000 most common words to give you a convenient base-10 system to work with. This grid would fit the entire Milky Way inside it while isolating Orion's Arm to a section of the grid system that is about 200x800x2000 sectors.
So your captain could say, "set course for sector cat-pop-bag" and with this simple 3-word combination, you establish one of a trillion possible, easy to say, sector names... and because they are alphabetical, they also give a good idea of displacement and distances so, even if you are outside of your usual territory, you would understand that "cat-pop-bag" is pretty close to "can-pond-back", but very far away from "door-zoo-gin". Over time, some of these sector names may even take on local pronunciations and be treated as words unto themselves like Catobag, Canonak, and Dozogin.
Also, because you are dropping some words, you could actually manipulate the word arrays just a bit do to something snarky, such as placing Earth in the Earth-Earth-Earth sector, or doing something practical like dropping offensive or overly similar sounding words.
Option B: Use local designations.
I remember back in the 90's (before cell phones messed everything up) if you asked someone for their phone number, they would give you a 7 digit number... but phone numbers were all 10 numbers long. This is because everyone would drop their area code because 99% of the time, you and the person you were talking to were in the same area code.
Now let's apply this same concept to a coordinate grid. Another way to break up the milky way into a trillion bite sized sectors is to break up this number into a series of 4 sets of 3 numbers where each set of numbers is an XYZ 10th of the previous subdivision such that your actual sector is a number between 000.000.000.000 and 999.999.999.999. So if you are in the sector representing the cardigan location X=1234, Y=5555, Z=6789 then your sector address would be 156.257.358.459. What this means is that if you are in sector 156.257.358.459, the closest sectors to you would be 156.257.358.359, 156.257.358.559, 156.257.358.449, 156.257.358.469, 156.257.358.458, etc... since everyone near by you is going to be in the same area code "156.257.358", then you in casual conversation can just say, "Set course for sector 458" and everyone knows that you mean the local sector 458 meaning 156.257.358.458. If you want to go a bit farther you might say "Go to sector 356 dot 458" meaning 156.257.356.458. The farther away you are talking about, the longer the number gets.
This way you can keep destinations simple small numbers most of the time. And ofcourse, over time people will have to come up with names for each of these levels of detail, like Domain, Zone, Section, and Sector so that if you want to reference a more broad area than a little cube of space containing no more than a handful of stars, you could say something like "The Moobark empire spans most of Domain 436". Now, you know you are talking about an area of space that is a billion sectors in size.