The technique used in most Lisp compilers (SBCL, etc) is as follows:
Say you have an outer lambda and an inner lambda. All variables (either arguments or local variables introduced by "let") defined in the outer lambda and used in the inner lambda are called the "free variables" of the inner lambda and the "closed over variables" of the outer lambda. Of the closed over variables of the outer lambda, all the ones that are assigned-to in the inner lambda are "assigned to variables." You can perform this classification in one pass before code generation.
Once you have annotated each lexical variable with its "closed over" and "assigned to" status, you can generate flat C functions in one code generation pass. The idea is to generate each inner lambda as a C function that accepts a hidden "environment vector" that contains the closed-over variables in the outer lambda. This environment vector is indexed simply by an integer index. Just handle the following AST constructs as follows:
LET: if the lexical variable introduced by the LET form is closed-over and assigned-to, it must be demoted to a box because the inner lambda can't modify the outer lambda's stack frame. Whereas you can generate a regular lexical variable directly as a C local variable, you have to generate a closed-over + assigned-to variable as a C variable that points to a single heap-allocated box that references a Lisp value.
VARIABLE-REF: for every reference to a variable, you have to look at it's status. If the variable is defined in the current lambda and not assigned-to, you can just emit a use of the C variable directly. If it is defined in the current lambda and assigned-to, you have to emit a load of the value from its box, a pointer to which is in a C variable in the current function. If the variable is defined in an outer lambda (it's a free variable of the current lambda), but not assigned-to, you have to emit the reference as a load (with a known constant index) from the environment vector. If it's a free variable and assigned-to, you have to load a reference to a box from the environment vector, then load the value from the box.
LAMBDA: assign each free variable of the lambda an index starting at 0 (which you will use when emitting references to free variables in the body of the lambda). Generate the LAMBDA expression as the creation of an object with two slots: a pointer to the C function of the inner lambda, and a pointer to an environment vector. Emit code to copy each free variable referenced in the inner lambda into the environment vector. Remember you might have multiple levels of nested lambdas, so follow the rules for VARIABLE-REF above when emitting each copy.
Then, just structure your calling convention so you can FUNCALL lambda objects and pas the environment vector as a hidden first argument.
Note that this requires a bit more environment book-keeping in your compiler than you might otherwise have. You really want each inner lambda to have a complete environment, including free variables. This is so you can keep track of which variable has which index in the environment vector. Say you have:
(defun foo()
(let ((x 0) (y 1) (z 2))
(labels ((bar () (+ x y))
(baz () (- z x)))
...)))
In "bar" your free variable indices probably look like {x: 0, y: 1} while in "baz" your free variable indices probably looks like {z: 0, x: 1}. You need to be able to keep track of these mappings separately, which you can't do if you attach these mappings in the environment structure you use for the LET form. Instead, you need to have a separate environment structure with entries in BAR and BAZ for the free variables of each lambda.
In practice this generates very good code, because most lambdas simply don't have very many free variables (at least so long as you don't naively generate LET forms as nested lambdas).