So far, Henne appears to be taking him up on the offer.
The 2008 second-round pick will try to lead the Jaguars to consecutive wins for the first time since 2010 on Sunday when they visit the sinking Buffalo Bills.
After replacing injured starter Blaine Gabbert in the first quarter at Houston on Nov. 18, Henne has completed 33 of 59 passes for 615 yards with six touchdowns and just one interception in two games.
Henne, who nearly led Jacksonville (2-9) to an upset win over the Texans, was 17 for 26 for 261 yards with two scores and one interception in Sunday's 24-19 victory over Tennessee.
With Henne providing a spark for the NFL's worst offense, the Jaguars have averaged 30.5 points and 389.5 total yards the past two weeks after averaging 14.1 and 263.6 over their first nine contests.
"He's made some good throws and some good reads for us," Mularkey said. "He, too, has things he can get better at in both games that he's played so far, but he's being very smart with the ball, especially in the pocket."
Cecil Shorts III and Justin Blackmon have benefited, combining for nine catches, 167 yards and two touchdowns on Sunday.
Shorts set a franchise record against the Titans with his fourth reception this season of at least 50 yards, surpassing the previous mark set by Keenan McCardell in 1998 and tied by Jimmy Smith the following year. Blackmon had a 7-catch, 236-yard performance in the overtime loss at Houston.
Tight end Marcedes Lewis has hauled in seven passes for 96 yards and two TDs with Henne under center.
"It just seems that everything is clicking right now,'' Blackmon said.
The Jaguars will likely have to rely on their passing game once again with Rashad Jennings averaging 2.8 yards per carry and Maurice Jones-Drew (foot) and Jalen Parmele (groin) out with injuries.
While Jacksonville could be in line for another big day against a Buffalo defense that has allowed 29.0 points and 380.5 yards per game, the Bills (4-7) have been better recently in holding Miami and Indianapolis to a combined 34 points and under 500 total yards.
After dropping its fourth in five games, 20-13 loss to the Colts on Sunday, Buffalo (4-7) desperately needs a win if it hopes to avoid extending a 12-year playoff absence.
"I don't really want to project or talk about the next however many games we have left," quarterback
Fitzpatrick was 17 of 33 for 180 yards with one TD and one interception against the Colts, while
Yet Buffalo's offense, which averaged 28.8 points in the first four games, has now topped 310 yards and scored at least 20 points just twice in its past seven.
The Bills have been especially inefficient in the red zone, scoring just nine touchdowns and 12 field goals in their last 25 drives. Buffalo has scored a TD on 45.7 percent of its trips to the red zone - 25th in the league.
While that could improve against Jacksonville, which ranks near the bottom of the NFL in points (28.0) and yards allowed (410.5), Buffalo will face a new-look defense with the addition of Jason Babin.
Babin, who was claimed off waivers from Philadelphia on Wednesday, was brought in to bolster a Jacksonville pass rush that has a league-low tying 13 sacks. The two-time Pro Bowl defensive end has 5.5 sacks after recording 18 for the Eagles in 2011-12.
"We're trying to win and we're doing whatever we can to improve our roster," Mularkey said.
The Jaguars will try for their third straight victory against the Bills. They won the last meeting in Buffalo, 36-26 on Oct. 10, 2010.