Lee Martin, 48, was taking part in an organised cycling event when he was hit by Christopher Gard's van on the A31 near Bentley.
Gard was jailed for nine years for causing death by dangerous driving.
The 30-year-old from Alton had at least six previous convictions for using a phone at the wheel.
After the trial, the Martin family criticised courts for failing to ban Gard from the road at prior hearings.
The victim's brother, Darrell Martin, said: "There were opportunities to stop the man from driving around.
"Just six weeks before he had persuaded a magistrate not to take his licence away and promised to lock his phone in the boot.
"The text message - think about how inane this is - it was about meeting his mate later and taking his dog for a walk. That's what killed my brother."
Police said Gard was texting immediately before the fatal crash on 12 August 2015.
Lee Martin, a father-of-two from Basingstoke, died later in hospital.
Gard was banned from driving for 14-and-a-half years when he was jailed on 5 September by a judge at Winchester Crown Court.
The Court of Appeal said Gard's application for leave to challenge his sentence had been passed to a "full hearing" on a date to be fixed,
Causing death by dangerous driving carries a maximum penalty of 14 years in prison.
A van driver who killed a cyclist after texting an "inane" message about a dog walk has been granted an appeal hearing against his sentence.