How we cite our quotes:
But every hour is saved
From that eternal silence, something more,
A bringer of new things; (26-28).
Earlier in the poem (20-21) Ulysses described the ways in which he could forestall death by continually moving; here he again refers to being "saved / From that eternal silence," but he doesn't tell us how it happens. Each "hour" that Ulysses is spared from death brings him "new things," new adventures, etc. He implies that the possibility for exploration is in the nature of life; whenever one doesn't die, "new adventures" or "something more" appear of their own accord.
And vile it were
For some three suns to store and hoard myself,
And this gray spirit yearning in desire
To follow knowledge like a sinking star,
Beyond the utmost bound of human thought (28-32)
In the same way that death and all the places that Ulysses wants to visit are conflated, so here the knowledge Ulysses seeks is combined with his own aspiration. "Like a sinking star" could refer to some fleeting knowledge Ulysses wants, or to himself! The suggestion is that the desire to explore "beyond the utmost bound of human thought" leads to death and makes one a "sinking star."
Come my friends
Tis not too late to seek a newer world (56-7)
"Newer world" is an interesting world; Ulysses might mean a world that is "new" to him, but there's also the suggestion that he's looking for a world that isn't as "old" as his is. Perhaps he means a world that isn't as primitive as Ithaca, where the citizens are no better than beasts; in this reading, Ulysses is attempting to find a more evolved place, one where he might feel more at home.