First of all, let me point out that it is really a rather difficult task for people who have not much experience in hymn playing. Playing a hymn with 2-3 lines of text between the staves is usually fine but when the space between the notes increases it really becomes similar to open score reading. The only difference from playing from an open score and a hymn written in this way is that in open score notation you have at least 4 staves with 4 parts and in hymn playing you have 2 staves.
By the way, perhaps practicing this way hymn playing is even more similar to reading open score because even in open score reading you are supposed to master solo parts, two parts at a time and various combinations of three parts before progressing to the complete four-part texture.
There are a few things which help me to play hymns written in this complicated way. This techniques may be useful to you as well:
1) My ability to harmonize the soprano line. You see, when I look at the hymn melody, in my mind I see the right chords which go well with the soprano part. In other words, I see the various chords (Tonic, Subdominant, Dominant, their inversions, seventh chords of various kinds etc.) which fit well with the melody. The skill at harmonization the melody allows me to naturally guess what notes will be written in the lower stave. Obviously there are usually many options available in harmonizing each particular measure but in many places, the most straightforward solution is the most common.
2) If I don't need to sing each verse while playing (this sometimes can be required, too), usually I just take a mental note of which verse I am playing without actually following the text. In this case I can concentrate on playing the music only.
3) Wise practice and experience also helps to overcome problems in hymn playing. I have a few hymnals at home which I have used to practice hymn playing earlier. I can say that after sight-reading about 100 hymns written this way, it gets easier and easier. I also recommend sight-reading hymns on a regular basis.
Remember that you don't need to play all 4 parts right away. This may well be what makes sight-reading a hymn a rather difficult task. If playing solo voices is too easy, practice (sight-read) in various combinations of 2 voices, 3 voices in a slow tempo. Only then play all the parts together.
Unless you struggle in note-reading, sight-reading separate parts might be a bit too easy because it is a simple chordal texture (except for the bass part in pedals). However, if you want to be really systematic like I am, you can play separate parts of these hymns as well.
Use the above tips today, open your hymnal and start practicing hymn playing the right way. Do this for only 15 minutes a day for some time regularly and you will start to notice some tremendous changes in your abilities.
By the way, if you really want to develop unbeatable sight-reading skills, check out my systematic Organ Sight-Reading Master Course which is intended for organists who want to perfect such seemingly supernatural abilities as playing fugues or any other advanced organ composition at sight. To successfully complete the practice material of this course will only take 15 minutes a day of regular and wise practice but you will learn to fluently sight-read any piece of organ music effortlessly.