"to go up the field a ways"

I found this phrase in some Chemical Brothers lyrics:

I thought we were going
To go up the field a ways
To join all the other living souls
But you never came

English's not my first language and I can't wrap my head around this. First, it looked grammatically incorrect to me, but a quick google landed me on a similar question here, discussing correctness of "a ways to go", but that was too unrelated.

I guess my question is: what does that line mean? I mean, without digging into interpretations of the song itself - I'm aware this is not a place to discuss meanings of lyrics. :) Or in other words, what would be a rephrased, grammatically correct alternative form, with the same semantics?

Thank you!

"a ways" is another way to say "an indeterminate distance". It usually includes a qualifier such as "quite a ways" for a longer distance or "for a short ways" for a lesser distance.

It can also be used to describe a length of time.

"Ways" is defined in the "Dictionary of Americanisms, Briticism, Canadianisms and Australianisms" by V. S. Matyushenkov

Meaning

"a ways" is a colloquial way to say a distance. It could mean any length of distance, so you need to use context to guess how much they mean.

You can say, "Go up the road a ways" to mean, "Go up the road an indeterminate (read: short) distance." You should guess this one is a short distance because if it were long, the person should warn you.

You can also say, "We have quite a ways to go" and that means, "We have a long way to go." Because "quite" intensifies things, we can guess that this is a long way.

Examples of use:

"Still a ways to go with online targeted ads"

"Marshall, Broxton remain a ways off"

Your Question

In the context of your song, you can probably guess. It's something like:

I thought we were going/ To go up the field a little bit /To join all the other living souls /But you never came

Use of "a ways" goes back many decades in U.S. usage. John Bartlett, Dictionary of Americanisms (1848) has this entry:

WAYS, for way, distance, space. A very common vulgarism. {It's only a little ways down to the village. —Margaret, p. 123.}

Likewise Frank Vizetelly, A Desk-Book of Errors in English (1906) says this:

ways, for way: In the sense of "space or distance," the erroneous form ways, for way, is often used colloquially, perhaps originally through confusion with the suffix -ways; as, "The church is a long ways from here," which should ne "The church is a long way," etc.

Modern commenters on grammar and usage avoid terms like "vulgar" and "erroneous," but you can still detect something less than hearty approval for "a ways" in comments like this one from Bryan Garner, Garner's Modern American Usage (2003):

way(s). In the sense "the length of a course or distance," way is the standard term {a long way}. Ways is dialectal. So it's surprising to find ways in serious journalism—e.g., "This is premature of course; Fox still has a ways to go [read some way to go?] before it's a full-fledged network." ... Newsweek, 6 June 1964, at 46.

Kenneth Wilson, The Columbia Guide to Standard American English (1993) has a fairly balanced (and, I think, accurate) take on the usage:

ways (n.) is the plural of way, and in many noun uses it is perfectly unexceptionable: There were three ways to get home. In some ways she seemed very poised and mature. Used instead of the singular, however, in phrases such as a little ways down the street, and as a suffix added to any, as in Anyways, I was too late, ways is Nonstandard, and many Standard users consider it a shibboleth inappropriate for any Standard use.

I understand "a ways" to be a modest distance. As in the following dialogue:

A: How far is it to the store?

B: It's just a ways up the road. Maybe a mile or two ahead.

The meaning is as in this answer the question you link to.