R plot Odds Ratio for each value in logistic regression

Question

I am trying to replicate the method in section 3a of this paper where, for a logistic regression model, they plot the functional relationship between a continuous variable and the odds for developing the outcome (using a smoothing spline).

I was wondering how do you replicate this from an R glm model, as example:

x <- c( 0,1,2,2,2,3,4,5,5,6,6,6,7,8,10 )
y <- c( 0,0,0,0,0,0,1,0,1,0,1,1,0,1,1 )

m <- glm( y ~ x , family = "binomial" )
summary(m)

Call:
glm(formula = y ~ x, family = "binomial")

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.6671  -0.5085  -0.3056   0.7814   1.6066  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)  
(Intercept)  -3.7311     1.9556  -1.908   0.0564 .
x             0.6906     0.3611   1.912   0.0559 .
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 20.19  on 14  degrees of freedom
Residual deviance: 13.47  on 13  degrees of freedom
AIC: 17.47

Number of Fisher Scoring iterations: 5

how can I extract the OR for each single value in order to replicate the example? Is the m$effect or what other parameter? I know how to plot predicted values, like this, but this is a different question