Teams that didn't finish ranked in their given year, due mostly to an unusually tough strength of schedule, but may have finished ranked in other years. A few candidates I considered were the 1998 team and the 2004 team, but I'll submit the 2013 team since I watched that entire season unlike the other 2 years I suggested. Would be good to hear from some of the older fans on older teams though.

The 2013 schedule was the toughest of any in BYU's independence era, ranked on average 43rd toughest in the nation, and third toughest in the post-LaVell era after the 2003 and 2004 seasons (sources: Sagarin, S&P, and teamrankings.com). The only other years the schedule cracked the top 60 in the post-LaVell era according to the average of the 3 sites I used are in 2012 and 2018.

What made the 2013 schedule so tough in particular was just the depth of the schedule. BYU played 7 P5 teams that year (4 of which finished with winning records, including 3 top 25 finishes), as well as 3 of the better G5 programs out there in Boise State, Houston, and Utah State. 9 of the 13 teams BYU played finished with winning records. That schedule was the closest BYU has ever had to playing a true P5 conference schedule, with only 1 FCS opponent. Probably the most similar schedule was in 2016 with 6 P5 teams as well as some good G5 teams in Boise State, Utah State, Cinci, and Toledo; although only 2 of those P5 teams finished with winning records, and 3 teams on BYU's schedule finished ranked that season.

You take that 2013 roster with Taysom Hill, Jamaal Williams, KVN, Bronson Kaufusi, Daniel Sorenson, Alani Fua, Michael Davis, Robertson Daniel, Uani Unga, Spencer Hadley, Paul Lasike, Kai Nacua, Dallin Leavitt, and even De'Ondre Wesley, and I believe they win 10+ games against most other of BYU's schedules and finish ranked in the top 25. They started off spotty in September, due to a brand new offense and perhaps the worst offensive line BYU has ever seen; peaked in the middle of the season against Utah State, GT, Houston, and Boise State during October; and injuries eventually caught up to this team, and they finished the season on a bit of a whimper.

The breakdown:

@Virginia, BYU loses 16-19 in one of the flukiest games I've ever seen. Multiple rain delays and installing Anae's brand new GFGH offense rendered this game completely rhythmless and out of sync. Still though, BYU was driving late to run out the clock, when a pass to Jamaal bounced off both his hands right to a waiting Cav player who, when getting tackled, then lateraled the ball to a open teammate to put UVA in prime scoring territory, allowing UVA to steal victory from the jaws of defeat. UVA finished an awful 2-10.

Texas, BYU wins 40-24. Sure Texas was overrated, but they finished 8-5 overall, including 7-2 in the Big12 which tied for 2nd in conference play that season. One of the best wins in the Independent era. It honestly deserves to be mentioned on any list of BYU's best wins throughout history, since it's not everyday BYU beats a P5 team with a 8+ wins by multiple scores, particularly one of college football royalty, a true blue blood.

Utah, BYU loses 13-20 in letdown fashion after the Texas win. That Hine KO return that got called back was awful. Perhaps it doesn't change the outcome of the game, but it certainly changed the complexion of it. For the most part though, Utah's defensive line had their way with BYU's offensive line. Utah only finished 5-7 that year, although they did steal a win over Stanford who finished #11 in the country and won the PAC12.

Mid Tennessee, BYU wins 37-10. Mid Tennessee finished 8-5, and 6-2 in C-USA. Not a pushover team that year as it turns out, yet BYU made quick work of them.

@Utah State, BYU wins 31-14, and the game was never really in doubt. Utah State finished 9-5 overall, including 7-1 in the MWC on their way to conference champions.

Georgia Tech, BYU wins 38-20. Georgia Tech would finish 7-6 overall, including 5-3 in the ACC, so another comfortable win over a P5 team with a winning record. Solid win.

@Houston, BYU wins 47-46 in exhilarating fashion on a TD pass in the closing seconds of the game. The game was Houston's homecoming game. Houston would finish 8-5, including 5-3 in the AAC to place 4th in their conference. Still a fun watch for any college football fan.

Boise State, BYU wins 37-20 in a game that wasn't even as close as the final score. Boise State had a down year by their standards, going 8-5 overall and 6-2 in conference to finish second in their division. Still, this was the first time BYU had beaten Boise State in football history, some still of some significance. This was also Chris Petersen's last year as the Boise State head coach.

@Wisconsin, BYU loses 17-27. The game wasn't particularly close in the second half, but BYU was simply outmatched this game, and IIRC this is the game when injuries really started to pile up. Wisconsin finished 9-4 overall, 6-2 in the B10, and was ranked #22 at the end of the season.

Idaho State, BYU wins 59-13. They went 1-11 in their lone season as an FBS independent, yikes. The only true gimme game on the schedule, unless you want to count Mid Tenn as well.

@ND, BYU loses 13-23. I thought this game would be closer, but BYU's offensive line was awful this game and gave no time nor room for Taysom and Jamaal. ND finished 9-4 and ranked #20 at the end of the season, although the wins got vacated at some point, as if that matters.

@Nevada, BYU wins 28-23. Game was closer than it should have been, given that Nevada only went 4-8, but BYU was pretty injury depleted by this point. Interesting nugget is that Nick Rolovich was the OC for Nevada this game.

*Washington, BYU loses 16-31 in a game that was never particularly close. BYU by this point had injuries all over the place, and simply did not have the depth to keep up. Washington finished the year 9-4 and ranked #25.