Multiple compilers in one Makefile - makefile

Question

I found this question here, which does exactly what I want. However, being new to makefiles, I am having a hard time seeing how to implement the second answer in my current setup.
I have a simple makefile for which I would like to use different compilers when compiling for Linux or cross-compiling for Windows. As presented, the file below works. To make for Linux I just type make, and for windows I type make os=win. In the interest of learning how makefiles work a little better, I would like to be able to implement the linked answer in the current Makefile so that I can make for windows just by typing make win or make cross as presented in the linked question.
Could someone help me understand how the structure presented in the linked answer could be implemented in the specific case I have here? Presumably there is a more elegant way than having native: and cross: targets and replicating almost the entirety of the code across both.
Any suggestions for cleaning up the Makefile in general are also welcome.
ifeq ($(os),win)
CC=x86_64-w64-mingw32-gcc
OUT=cusum.exe
else
CC=gcc
OUT=cusum
endif
CFLAGS=-D_GNU_SOURCE -O3 -Wall -Wextra -lm --static
DEPS=bessel.h detector.h io.h stepfit.h lmmin_int64.h utils.h
ODIR=obj
_OBJ=main.o bessel.o detector.o io.o lmmin_int64.o stepfit.o utils.o
OBJ = $(patsubst %,$(ODIR)/%,$(_OBJ))
LIBS=-lm
$(ODIR)/%.o: %.c $(DEPS)
$(CC) -c -o $# $< $(CFLAGS)
$(OUT): $(OBJ)
$(CC) -o $# $^ $(CFLAGS)
.PHONY: clean
clean:
rm -f $(OUT) $(ODIR)/*.o *~ core $(INCDIR)/*~

Answer 1

You only have to introduce an upper-level target. Target-specific variables are inherited by their prerequisites, so you don't have to change all the targets.
However, you cannot update the content of targets or prerequisites using this method so you can't remove the assignment of OUT. This means you can't get rid of the OS variable assignment.
One simple way to do it is to use recusive invocations of make. Write your makefile normally:
O = o
E =
CC = gcc
OUT = cusum$E
CFLAGS = -D_GNU_SOURCE -O3 -Wall -Wextra -lm --static
DEPS = bessel.h detector.h io.h stepfit.h lmmin_int64.h utils.h
ODIR = obj
_OBJ = main.$O bessel.$O detector.$O io.$O lmmin_int64.$O stepfit.$O utils.$O
OBJ = $(patsubst %,$(ODIR)/%,$(_OBJ))
LIBS = -lm
$(ODIR)/%.$O: %.c $(DEPS)
$(CC) -c -o $# $< $(CFLAGS)
$(OUT): $(OBJ)
$(CC) -o $# $^ $(CFLAGS)
.PHONY: clean
clean:
rm -rf $(OUT)* obj wobj *~ core $(INCDIR)/*~
Then add a target that recursively invokes make overriding the relevant variables (be sure to put it at the end or at least after the $(OUT) target):
win:
$(MAKE) CC=x86_64-w64-mingw32-gcc E=.exe O=obj ODIR=wobj
Now you can run make win and it will recursively invoke make overriding the appropriate variables.