pasting into a filtered list

hi im having trouble with excel like, everyone here!

i have a list of names in excel, which i can filter using advanced filter based on the criteria of another list, i want to paste the data of the second list next to the names in the first list when its filtered. but when you paste next to a filtered list it puts the data into the hidden cells.

how can i paste only into the visable cells, without having to copy and paste the filtered lists somewhere else?

e.g. first list:
aaa 123
aba 12
bba 31
aab 325
bba 234
bbb 63
abb 326
bab 324

second list:
aba 124
bba 125
abb 634
bab 436

so what i want in the end looks like this:

aaa 123
aba 12 124
bba 31 125
aab 325
bba 234
bbb 63
abb 326 634
bab 324 436

i know that it would be easy with this list as it is so small but i have thousands of rows, and hundreds of columns. does this make any sense?

thankyou very much to someone who can help me!!

thanks a lot martin, im not sure how to use vlookup, but i think i might be able to do it the other way you mentioned.

another problem ive just come across is that when filtering the data using the advanced filter it searches for data which CONTAINS words from the criterior, is there any which i can make it only show EXACT MATCHES.

i.e.

aaa left
aca right
ads
afs
afe left
cve

is the criterior list is

aaa
aca
ads
afs
afe right
cve

then advanced filter will show:

aaa left <---- this is not technically in the criterior list
aca right <---- this is not technically in the criterior list
ads
afs
cve

does this make sense?

thankyou for your time

ok thank a lot for your help i think ive solved it now.

if anyone else is having a similar problem, i used find and replace and replaced the space with a random letter which is not in either of the listS, then switched the letter adjecent and to the left with this new digit, so that it would not come up in any of the searches unless it was an exact match

for example

AAA LEFT

becomes

AABALEFT

then once the whole list is compiled i can replace 'BA' with 'A '

thankyou for all your help!