The fact that you have at least one hyphen in the phrase means you’re already on the right track. First, let’s back up and look at why you need hyphens in a phrase like this.
The phrase “ten-day trip” includes a compound modifier, a group of words (ten-day) that modify a noun (trip). The individual words in compound modifiers rely on one another. If one stood alone with the noun, the sentence probably wouldn’t make sense or would change in meaning. That certainly applies here:
I’m going on a ten trip.
I’m going on a day trip.
Using a hyphen with compound modifiers that come before a noun clarifies meaning and makes the sentence easier to read.
You’re asking about a phrase that’s more complicated. It’s essentially saying this:
I’m going on either a six-day trip or a ten-day trip.
But that’s wordy. So, the choice to pare down is a good one. In cases like this, use a suspended hyphen to indicate each modifier’s connection to the noun:
I’m going on a six- or ten-day trip.
Make sure you include a space after the hyphen for the first modifier so it’s clearly suspended and not meant to connect two adjacent words.