First off, a hiring manager wants to know what you're doing now -- it will help her assess whether you have the right skills to succeed in the new role. Hopefully, your most recent position is a natural lead-in to the job you’re applying for, but even if it’s not, it’s still important for hiring managers to know what you’ve been doing most recently. Think of skills like a muscle -- the ones you exercised at your latest position are the ones in the best shape. If the job you’re applying for is in a different field than your most recent job, choose transferable skills to highlight.
If your resume lists jobs in a random order (or divides experience into two buckets, “relevant” and “other”), you’ll confuse hiring managers and tell an incoherent story. Even if you think you’re showcasing why you’re right for the job by listing what you deem your best experience first and burying the odd-jobs that helped you pay the bills, you’re actually forcing the hiring manager to think harder than she’d like to and put together a puzzle of your work timeline. Imagine a tired hiring manager reading your resume and thinking, “Ok, Jane Doe’s last job was in 2013 in LA...but then in 2016 she also had a job working remotely for a company in London? Hmm, from 2014 to 2016 she lived in Toledo…where is she now?” Does this person want to do the mental gymnastics to figure out who you are and what you’re doing? No. She wants to move on to a candidate whose resume is clear.
If you do need to explain any gaps, inconsistencies, or extenuating circumstances in your resume, do so in your cover letter. When you have full sentences and paragraphs at your disposal, it’s easier to weave together your story in the chronology you’d like. But a resume story is supposed to be clear, concise, and chronological.