Some school districts do have longevity increases in their stipends for coaches ( so the more years you are at the school the more you get paid). Additionally, some school district do give additional supplements for each week that that they advance in the playoffs (but a super majority of them do not).
Where is it written that a booster club can't give money to a coach for off-season programs? For running a 7 on 7 team, or running the weight room in the summer time.
To answer the original question though... yes, coaching for almost all sports has become an all year function. The days of part-time coaches is slowly vanishing. You still see part time coaches in sports where the school can't get someone who wants to do fulltime or they have ulterior motivies. A friend applied for a girls basketball position at doormat school (he was "qualified" having been an assistant coach for an above average program for 5-6 years). The school instead gave the position to an assistant football coach so that they could supplement his income a little more because they knew he wasn't getting much as a football coach. The school is still a doormat program (I don't think they have won more than 5 games in any year in the past 20 years).