By 2012, the Steven Moffat/Matt Smith-era of Doctor Who was well into its third season (“series”).
The show had gone through not one, not two, but three different costume designers before Howard Burden finally took over at the beginning of season 7. Even then, he generally adhered to Matt Smith’s previously-established look for the first several episodes (now typically regarded as “season 7a”).
In the interview below, he said that he was told partway into the season that he could actually change the “look” of the character:
It was in the 2012 Christmas special, “The Snowmen,” that Eleven’s velvet waistcoat first appeared, as a component in his “Snowmen” costume ensemble.
While I don’t know if this “Snowmen” costume was a prototype and/or experiment for a new “season 7b” costume, or if it was always intended as a “one-off” costume for that particular Christmas special, what we do know is that it was a very different style costume than those the 11th Doctor had previously worn.
However, this costume analysis isn’t about the “Snowmen” costume in its entirety – only the velvet waistcoat with which it appeared.
He wore the velvet waistcoat through the entirety of that special, as well as in its corresponding publicity photos.
These included this particular costume variant, from when the Doctor was cosplaying as Sherlock Holmes:
After that special, though, the Doctor’s velvet waistcoat wouldn’t be seen again until the penultimate episode of season 7, “Nightmare in Silver.” (In the interim, he’d either worn his “scales” waistcoat, or no waistcoat at all.)
However, “Nightmare in Silver” was significant not only because of the velvet waistcoat’s return, but because it was the first time he’d worn it with his “7b” costume, as opposed to his “Snowmen” costume.
He also wore it in the publicity photos for the episode.
The Doctor wore the velvet waistcoat in the short prequel (?) to the season 7 finale, “She Said, He Said.”
This made a degree of sense, as he continued to wear his velvet waistcoat in the actual season finale, “The Name of the Doctor.”
He also wore the velvet waistcoat in the publicity photos for that episode.
He only wore the waistcoat for a little while this time, shedding it once we began “skipping ahead” and saw long periods of time passing.
At some point, he apparently began wearing the waistcoat again, though, since he was wearing it near the end of the special – his whole costume having been distressed to represent long centuries of “wear and tear.”
He also wore the velvet waistcoat with his “Snowmen” costume in many publicity photos for the special.
Matt Smith was wearing the velvet waistcoat during some behind-the-scenes specials on the season 7 Blu-Rays, presumably quick bits filmed all in one go between filming of “Nightmare in Silver.”