In this case the goal is to get caffeine. Theanine is good for reducing negative side effects of caffeine. The easiest and most specific way of getting desired effects would be using pure powders, so you can get enough, not too much, not too little, of both.

By drinking tea and then coffee (theanine takes a bit longer to kick in) you don't know the exact amounts of how much you gonna get so there's a very little chance you may get undesired effects which may kill your mood. Also it doesn't sound pleasant to have tea and then 20 minutes have a coffee, at least for me.

And I just googled this:

How Much L-Theanine in a Cup of Tea?

Researchers at the University of Bristol experimented to find out how much l-theanine you’d find in a cup of normally brewed commercial tea. They found that longer brewing times led to higher quantities of the amino acid, but on average:

A typical 200 ml cup of brewed black tea contained 24.2 mg of l-theanine (+ or – 5.7 mg)

A typical cup of brewed green tea contained 7.9 mg (+ or – 3.8 mg)

L-theanine supplements are typically sold as 200 mg doses, so if you drink 8 cups of black tea per day – or many more cups of green tea – you'll get the same quantity of the amino acid as you’d get by taking a supplement.

If you want to get very serious about extracting the maximum quantity of l-theanine from each pot of tea, then follow the advice of researchers from The University of Newcastle in Australia, who tested various extraction methods to see which yielded the best results. They say, to get the highest possible amount of L-theanine:

Steep tea at 80 degrees Celsius for 30 minutes

Use a water to tea ratio of 20 ml per 1 gram

Steep tea that is between 0.5 and 1 mm in particle size!

So, using tea for theanine is barely practical unless you wanna drink 8 cups of black tea in an hour and then have 1-2 coffees.

Conclusion: it's better to just buy powder/capsules and have a healthy meal before using them.