dry eye (keratoconjunctivits sicca (KCS) causes eyes to be red but often, not always, has discharge associated. There are tests any vet can do to check for this a Schirmer tear test, it's not too expensive to have done and as far as I know any vet can do it (it's pretty simple).
I could see it getting worse later in the day cause their eyes have been open longer, just like people, after some "shut eye", tear production builds up and things aren't so bad. My vet told me to watch for red eye whites as a number one sign of this.
Something simple that you could rule out.... and fairly easy to treat. There are long term complications with untreated dry eye, incl. blindness.
and Hazels eye drop suggestion will help, though there are other ointments (cyclosporine - optimune) that treat dry eye specifically
Magnolia (aka Nolie) - 6 yrs old, adopted May 3, 2009 - mixed breed monster mutt
R.I.P. Chloe - my first Golden girl - we said goodbye but you will always be here (1994-2007 adopted April 2000)
R.I.P. Molly - my Golden girl, who's causing havoc over the bridge now - (1999-2009 adopted May 2007)
and love for my "lost ones" - Misery, Clover & the girls (chickens)