How can I make a splice in a 30 amp 240 circuit?
I am a huge fan of steel boxes for things like this for a couple of reasons.  First, they contain arc faults very well, they won't melt or burn through like plastic, and if anything will help the breaker trip if a hot wire comes loose.    Second, they solve a problem the last guy had: finding a proper cover for the box.  Blank cover plates are perfectly common in metal boxes. 

I would use 4" x 4" x 2-1/8" (deep) box and lid for the splices. You will also need cable clamps that fit into knockout holes, a bag of enough of those should be $2-ish. 

The constraint on box size is cubic inches required space: in each box you need 5 cubic inches plus 7.5 cubic inches per cable. That's 20 cubic inches for a 2-splice and 27.5 for a 3-splice.   That box is 30.3 cubic inches. You can get 7.0 more with a domed cover. 

Lastly you will need some #10-32 ground screws and two 8" lengths of #10 bare wire (which can be cut down from spare 10/3 cable, e.g. if you scrap the current middle section). 

Position the boxes

One hole in the boxes will be threaded #10-32 for a ground screw. It may be on a little dome.  Otherwise be careful to position the box so the ground screw is usable.  Put an 8" ground pigtail on that screw. 

The key is to position junction boxes in locations that are a) properly mounted, b) allow at least 7" of wire ends to enter the box; and c) allow routing of the dryer and supply in a neat and workmanlike manner compatible with any future plan you might have with the use of the space. 

You don't need to worry about having that short jumper reach; we can easily replace it.   You also don't need to worry about the garage run reaching either of these box positions (It might be nice, but don't sacrifice a good box placement to get it).  

Boxes can be attached to rafters, or you can lay a 4" wide sheet of plywood between rafters if you want a particular position.  

Linking

After the junction boxes are mounted, attach the cables via knockout holes and cable clamps. The cable clamps are mandatory, don't just bring it in a hole.  

Feel free to reuse any wires in good shape.  Install the run between the boxes.  Then also install the garage run.  If it's not long enough, either replace the whole run or add a third junction box to splice and extend it.  

There is nothing wrong with wire-nut splices if they're properly done.  For #10 wire, that means use the big nuts, line them up properly, and twist gorilla-tight.  Then do a very firm pull test (hold nut, pull each wire in turn). There is no way a #10 will come out (if done right). 

Nut the junction box's ground wire in with the other ground wires. This is cheap insurance in case a hot wire ever gets loose.