The problem with your no-splicing plans is the cost of copper
While avoiding splices, boxes, and access points sounds like a good idea, the problem with your plan is that copper 3/3/3/5 SER cable is quite expensive stuff at over $5/ft yet only a hair smaller at 0.91" diameter vs 1.08" diameter for 1/1/1/3 Al. This means that you're looking at a few hundred dollars of SER cable alone for either your plan 3 or the plan 4 you mentioned in your comment, and makes the parts required for a splice rather cost-effective in comparison. Going to 1/1/1/3 aluminum SER at about $1.60-$1.70/ft makes avoiding splicing a cost-effective option, though, and in that vein, I would favor plan 4 with cutting the existing feeder and rerouting it, then replacing the rest of the existing feeder run with a new run. (Aluminum cable is also lighter than copper cables with the same ampacity, which factors into the ease of pulling significantly at the larger gauges that aluminum wire is typically used for.)
If you do go that route, you'll want to use a subfeed lug block in the new subpanel to feed the existing panel instead of a 100A breaker since the breaker would be fairly pointless anyway due to the inability to coordinate it with the existing feeder breaker. If you really want a disconnect for the original subpanel in the new subpanel, then you can use a 100A molded case switch of the appropriate type for the new subpanel.
If you are to splice....
If you do go with the splice-box plans, there are a few points you will need to keep in mind. First off, the T-splice is a much better idea (less cable and fewer splices) than the "loop" splice, so we'd be going with your plan 1 in this case instead of your plan 2. Furthermore, you will need to have the splice box be accessible now and forevermore; you can paint or otherwise finish the cover to match the surroundings, but actually burying a box is no good. Finally, we'll have to account for the NEC 300.14 requirement of 6" minimum of free conductor for each cable entering the box; if you can't get that from slack in the existing run, you'll have to go up to a 12" long by 6" to 8" wide by 4" deep pull box and add an extra block and some 2AWG copper jumpers to connect the two, as well as a spare grounding lug such as a Brumall 6T attached to the box with a 10-32 self-tapping/self-drilling grounding screw such as the Garvin GSST.
Getting into the details of making the splice, we'll first need a big box, far larger than the junction boxes you're used to, due to NEC 314.28(A). In particular, you'll need an absolute minimum of a 6" long by 8" wide by 3" or 4" deep, NEMA 1 (indoor) pull box with a flush-mount cover, grounding kit, and 8AWG grounding jumper in order to accommodate such a tap, using dual-rated (AlxCu) insulated mechanical splicing connectors (Polaris™ or equivalent). However, it may be cheaper to use a larger box so that you can use a UL listed (UL1953/QPQS) power distribution block in accordance with NEMA 314.28(E) as Polaris-type connectors tend to be rather expensive compared to distribution blocks, and also lead to a somewhat messier install compared to the neatness a distribution block can provide in this application.
If you do decide to go the distribution block route, you'll need an 8" long by 8" wide by 4" deep NEMA 1 pull box with a flush cover, a dual-rated (AlxCu) 3-pole power distribution block such as the MPDB63193 to make the splices, a few appropriately sized sheet-metal screws to mount the PDB to the box, and some grounding parts. In your case, I would use an Ilsco NBAE-0307-1 mounted with a 1" 10-32 machine screw through the supplied grounding hole in the box to attach the ground wires to the box. If the box lacks a grounding hole, you can use a 1" 10-32 thread-rolling screw (Taptite™ or equivalent) into an appropriate field-drilled pilot hole instead of the 10-32 machine screw.
Either way, you'll need appropriate bushings and SER clamps to get the cables into the box. You may also need to make your own KOs in the sides of the pull box using a punch set in order to fit the SER clamps in the appropriate places so that you have sufficient slack in the existing wires to make your connections. Some pull boxes ship without knockout rings pre-stamped in them to begin with, even!
Go big or go home!
If you don't already have the new panel installed, I would get the biggest panel I could in order to save you the hassle of having to upgrade it down the line. In particular, a 24- or 30-space, 125A, main lug panel would not be at all out of place here. If you feel like putting the money you're saving on wire to work, you can even use a 40-space or 42-space, 200A or 225A main lug panel for the new subpanel. Either way, you'll need to fit the panel with separate grounding bars and pull the bonding screw from it, just like any other subpanel.
TORQUE ALL LUGS TO SPEC
Last but not least, you'll want to use an inch-pound torque wrench to torque all the various connection setscrews to their labeled torque specifications. This is a new requirement in NEC 110.14(D) as of the 2017 NEC, and a good idea anyway in order to ensure that your connections will pass the test of time.