The Prison Reform Trust has published a report calling for inmates to be encouraged to volunteer and for "active citizenship" to be promoted behind bars.
The charity said the move would help make "prisons places of hard work and purposeful activity".
Justice Secretary Ken Clarke has previously said many prisoners are living a life of "enforced, bored idleness".
The report, Time Well Spent, found that overall volunteering is open to very few prisoners.
Prison Reform Trust chairman Lord Woolf said: "The best way of avoiding those heart-rending suicides, particularly with young people while they are in prison, is to motivate other prisoners to take responsibility."
David Ahern, CEO of the Shannon Trust, whose Toe by Toe programme sees more able prisoners teach others to read and write, said: "We know that going through the prison system there are around 50,000 people each year with a reading age of that expected of an 11-year-old or below.
"Now we are engaging with a few thousand.
"There is this huge swathe of people in prison who we could be helping that we are not helping."