It's more that it is after the conjugated verb since it will be taken as modifying "veux". http://french.about.com/library/weekly/aa060300.htm
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The main difference with English is that we cannot place an adverb between the subject and its conjugated verb.
The second difference is that (much like adjectives) our adverbs modify the preceding word and not the following word.
Keeping that in mind, if I want to align the action to someone else's:
- moi aussi, je veux savoir
If I want to combine 2 modals:
- je peux savoir et je veux aussi savoir
If I want to combine 2 actions:
- je veux comprendre et je veux savoir aussi
But this is theoretical and context should shed light on what is associated to what.
"I would like to know, too!" should be accepted. "Too" seems much more natural to me (native english speaker). Reference: https://dictionary.cambridge.org/us/grammar/british-grammar/also-as-well-or-too