My general time frame for starting seasons is 4-6 hours leveling (I suck at leveling). After reaching 70, spend 1-2 hours to reach T4. After reaching T4, spend maybe 2 hours or so to reach T6 depending on RNG. I was running T6 by the weekend of S2 launch. Even if you don't play often, 60 hours to reach solo capable T6 is a bit slow. So as far as season progression goes IMO, you should hit 70>T1>T4>T6.
Do all of your basic preparations in T1, then get out of it as soon as possible. Gather rare equipment and gems. Make sure your rares are properly itemized. The biggest mistake I see people make early in the seasons is gearing with non-BIS and badly itemized legendaries so early on. It would only be worth using a non-BIS leg if it doesn't have to be enchanted (like Goldskin that rolled with Stat/Vit/3Socket). Besides that, you're better off crafting rare gear with proper stat itemization like trifecta gloves. Only do enough T1 rifts, to get a couple keystones of trials. From my experience, slow rolling through low level GRs net more legendaries than actual T1 runs and you'll at least be able to collect all of your legendary gems. An unleveled Bane of the Powerful and even Zei's are instantly a big boost.
To further streamline the gearing process, use a Broken Crown to collect gems to craft rare rings and amulet. Getting a Stat/CC/CHD/Socket ring and amulet as early as possible is a big DPS boost. If you were lucky to get a RoRG from bounties while leveling, you'd only have to craft one ring (to be eventually replaced by an SOJ or maybe a Unity). Itemized legendary amulets are hard to come by so crafting one is more than worth it. Also get Boon of the Hoarder early so gold isn't an issue.
You can hit T3-T4 once all of your gear are properly itemized whether they are legendary or not (aim for 400k+ sheet DPS). T4 is a bit better for getting legendaries than slow rolling through GRs. Its pretty much mostly RNG at this point. Hopefully you'll have obtained some set pieces while running GRs. Getting a 4 set for classes like Barbs, DH, and Monks will boost them to T6 really quickly. Getting a 6 set should more than be enough to solidly farm T6.
To answer your question, there isn't a whole lot of in-betweens as far as efficiency goes. Clearing T6 rifts in less than 8 minutes is good enough at low paragon levels early in the season, but you should be closer to 5 minutes or less once you have your BIS pieces itemized.