The art of resteeping?

Also… there is Gaiwan/gongfu brewing, used mainly with Asian unflavoured teas. You can use any smaller vessel really, but the traditional way is using this type of cup for steeping:
https://encrypted-tbn3.google.com/images?q=tbn:ANd9GcQK3Rh1qcXDEnDk-OgUKgGs0zeTMK-R3lyaUy9RSkTF79YHls5p
The basic rule of thumb here is that the more tea you use in the Gaiwan, the less time you allow it to steep for, ie as short as 10 seconds, or you need to use more water (ie. a larger cup)
This is the Gongfu option. Very similar in concept:
http://www.google.ca/imgres?um=1&hl=en&client=firefox-a&rls=org.mozilla:en-US:official&tbm=isch&tbnid=bIuOSYq_54KO_M:&imgrefurl=http://www.stoneleaftea.com/products-page/all-inclusive-tea-sets/gong-fu-tea-set/&docid=Si_NwNjZv_XhwM&imgurl=http://www.stoneleaftea.com/wp-content/uploads/2011/11/gongfu-set.jpg&w=800&h=638&ei=D-8yUPBtsY3oAbbLgNAB&zoom=1&iact=hc&vpx=681&vpy=167&dur=894&hovh=200&hovw=251&tx=161&ty=105&sig=117745789684156571129&page=1&tbnh=124&tbnw=162&start=0&ndsp=19&ved=1t:429,r:9,s:0,i:101&biw=1366&bih=609
Something to consider anyhow. As long as you have a cup/infuser method of a similar size, you can try this method.
One other thing to note is that with each infusion you should increase the steep time. Different people have unique preferences but mine is to raise the time by about 25% or in 1 minute increments.

Libertea has given lots of helpful pointers, just adding on a little.

This is a fun topic and helpful in encouraging people to try quality tea.

For example well-made oolong teas can be steeped anywhere from 4-10 times. So you may pay maybe twice as much but you can steep 3 times as much as a mediocre oolong so on a per cup basis, it’s makes more economic sense.
Here are some of my experiences on this topic:
i) Gongfu brewing is most appropriate- the combination of smaller utensils, higher tea leaves ratios and shorter steeping times help you to
extract the full flavor of the tea and more steepings

ii) Unflavored tea works better- few herbs/spices/fruits/additives stand up to as many infusions as tea, in the beginning, these flavorings may be obvious but by the 3rd or 4th the flavor is gone. (Incidentally this is a good way to discern genuine dancong from the fakes)

iii) Here are some general parameters that may help
Type: Infusions Additional steeping time
Green 2-4 30 secs to 1 min
White 3-4 30 secs to 1 min
Yellow 3-4 30 secs to 1 min
Black 2-4 1 min
Oolong 4-10 15 sec to 45 sec
Dark tea (Pu-er) Wide range 30 sec
Some people can enjoy more than 10 steeps of dark tea, depends on the type really. Some of it gets quite bland after the 5th.

iv) A note for tightly rolled/compressed tea such as Pu-er in tuocha or discus form (bing cha), between the 1-3 steeps you may not need to add the infusion time because the more ‘unfurled’ these compressed tea are, the more flavor they release and it mitigates the loss in flavor. In fact, for compressed tea, often I like to reduce the steeping time

v) You may want to keep a small cup of each infusion for comparison purpose, I used to do this to test a hypothesis and adjusted my steeping time to make the liquor color of each steeping constant. (okay, i am a bit more of an extremist because I kept a log and took a photo)

vi) Let your nose guide you, especially when using a gaiwan. You can roughly tell if the tea is steeped enough from the aroma.
Also, by smelling the wet leaves, if there is still some fragrance in the tea leaves, it probably means you can still continue steeping it.

Bottom line: Yes, do re-steep. You may find the flavor changes from steep to steep and each infusion is a new experience.

Hi there…
My 2 cents worth…

A finum little basket is handy for resteeping simply in a mug. It’s cheap and easy.

Since you ordered from Verdant, go to the website and follow the instructions for steeping times. Some of the tea’s say to add as little as 5 seconds for additional steepings. If a vendor has instructions, it means they’ve done the work already to find the best flavor. I always try their way before I venture out on my own. (Prevents waste of precious tea)
When I’m sloshing, I keep steeping sometimes and put the tea into the frig to cool for drinking later. No waste there either.