Meaning of "To be next into the breach"

According to thefreedictionary.com, "step into the breach" means "to do someone's work when they are suddenly not able to do it [eg] Professor Collier stepped into the breach when the guest lecturer failed to turn up." As shown at phrases.org.uk (and noted in previous answer) the phrase stems from the 'Cry God for Harry, England, and Saint George!' speech of Shakespeare's Henry V, Act III. This link suggests the meaning is "Let us try again one more time", rather than doing something no one else will do.

Breach means ‘gap’, and the phrase is an echo of

Once more unto the breach, dear friends, once more

from Shakespeare’s ‘Henry V’. There the breach is the gap in the French defences made by the English army.