Forth (gforth), 81 bytes

: f begin 1+ >r i i 3 mod 1 r> begin 10 /mod >r 3 <> * r> ?dup 0= until * until ;

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###Explanation

1. Add 1
2. check if multiple of 3
3. check if contains a 3
4. if the result of step 3 or step 4 is true, repeat from step 1

###Code Explanation

: f               \ start a new word definition
  begin           \ start an indefinite loop
    1+            \ add 1 to current number
    >r i i        \ put current number on return stack, then place on normal stack twice
    3 mod         \ check if number is multiple of 3
    1 r>          \ place a 1 on the stack, then move the current number to the normal stack
    begin         \ start another indefinite loop
      10 /mod     \ get quotient and remainder of dividing by 10
      >r          \ put quotient on return stack
      3 <>        \ check if remainder does not equal 3
      *           \ multiply by accumulator (cheaper version of "and") 
      r>          \ remove quotient from return stack
      ?dup 0=     \ if quotient does not equal 0, duplicate, else put -1 on stack
     until        \ end loop if top of stack does not equal 0
     *            \ multiply result of both checks (cheaper "and")
   until          \ end outer loop
 ;                \ end word definition

Forth (gforth), 81 bytes

: f begin 1+ >r i i 3 mod 1 r> begin 10 /mod >r 3 <> * r> ?dup 0= until * until ;

Try it online!

Explanation

1. Add 1
2. check if multiple of 3
3. check if contains a 3
4. if the result of step 3 or step 4 is true, repeat from step 1

Code Explanation

: f               \ start a new word definition
  begin           \ start an indefinite loop
    1+            \ add 1 to current number
    >r i i        \ put current number on return stack, then place on normal stack twice
    3 mod         \ check if number is multiple of 3
    1 r>          \ place a 1 on the stack, then move the current number to the normal stack
    begin         \ start another indefinite loop
      10 /mod     \ get quotient and remainder of dividing by 10
      >r          \ put quotient on return stack
      3 <>        \ check if remainder does not equal 3
      *           \ multiply by accumulator (cheaper version of "and") 
      r>          \ remove quotient from return stack
      ?dup 0=     \ if quotient does not equal 0, duplicate, else put -1 on stack
     until        \ end loop if top of stack does not equal 0
     *            \ multiply result of both checks (cheaper "and")
   until          \ end outer loop
 ;                \ end word definition