There are a lot of different things that can happen in harmony(sometimes called mechanisms). It's a lot easier to write or even improvise parts if you know which one (or ones) you want to use.

First, there is parallel motion, which often would simply be a second line that follows the melody exactly, but, for instance, a third/minor third apart-(you could do any interval, but most wouldn't work well in a folk arrangement).

Next, there is similar motion, where a second line follows the melody approximately, but doesn't always keep the same interval separation--This tends to change pitch less than the melody--often only when a chord change occurs. "Grassroots" harmony often uses this--

Then, there are the various species of counterpoint, where one or more voices move in different directions from the melody. Here's where we fit in the chord arpeggiation that Artful Codger mentioned.

Some harmony is based on creating a chord with the melody note on top, other times, block chords are sung with the melody floating independently.

Old Style R&B used(and uses) the melody line floating over chordal rhythm patterns.

The old 50's early 60's pop/folk groups might have done an arrangement like this:

A couple lines (or even a verse) in unison, then a couple lines to a verse in two part harmony, followed by the refrain starting in two part harmony leading in to a full four voiced chord. Then, on either a bridge or a third verse, there might be a solo while the other voices hum, "lalala", or "Waaaahh" on a a block chord. Then back to unison for the last verse.

When you've made up your mind what you're going to do, it becomes a lot easier.